3.11.94 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^2} \, dx\)
Optimal. Leaf size=292 \[ -\frac {d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (7 A b e-4 A c d+2 b B d)}{b^3}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac {e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 b^2 c^2}+\frac {(c d-b e)^{5/2} \left (-3 A b c e-4 A c^2 d+5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}+\frac {e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{b^2 c^3} \]
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Rubi [A] time = 0.88, antiderivative size = 292, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used =
{818, 824, 826, 1166, 208} \begin {gather*} \frac {e \sqrt {d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{b^2 c^3}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac {e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 b^2 c^2}+\frac {(c d-b e)^{5/2} \left (-3 A b c e-4 A c^2 d+5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}-\frac {d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (7 A b e-4 A c d+2 b B d)}{b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x]
[Out]
(e*(2*A*c^3*d^2 - 5*b^3*B*e^2 - b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(8*B*d + 3*A*e))*Sqrt[d + e*x])/(b^2*c^3) + (e
*(6*A*c^2*d + 5*b^2*B*e - 3*b*c*(B*d + A*e))*(d + e*x)^(3/2))/(3*b^2*c^2) - ((d + e*x)^(5/2)*(A*b*c*d + (2*A*c
^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*(b*x + c*x^2)) - (d^(5/2)*(2*b*B*d - 4*A*c*d + 7*A*b*e)*ArcTanh[S
qrt[d + e*x]/Sqrt[d]])/b^3 + ((c*d - b*e)^(5/2)*(2*b*B*c*d - 4*A*c^2*d + 5*b^2*B*e - 3*A*b*c*e)*ArcTanh[(Sqrt[
c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(7/2))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 818
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Rule 824
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]
Rule 826
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Rule 1166
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} c d (2 b B d-4 A c d+7 A b e)+\frac {1}{2} e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} c^2 d^2 (2 b B d-4 A c d+7 A b e)+\frac {1}{2} e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{2} c^3 d^3 (2 b B d-4 A c d+7 A b e)-\frac {1}{2} e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^3}\\ &=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} c^3 d^3 e (2 b B d-4 A c d+7 A b e)+\frac {1}{2} d e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 A e)\right )-\frac {1}{2} e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^2 c^3}\\ &=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\left (c d^3 (2 b B d-4 A c d+7 A b e)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b^3}+\frac {\left ((c d-b e)^3 \left (4 A c^2 d-5 b^2 B e-b c (2 B d-3 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b^3 c^3}\\ &=\frac {e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 c^2}-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac {d^{5/2} (2 b B d-4 A c d+7 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {(c d-b e)^{5/2} \left (2 b B c d-4 A c^2 d+5 b^2 B e-3 A b c e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.55, size = 341, normalized size = 1.17 \begin {gather*} -\frac {-\frac {105 \left (-2 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\frac {2}{15} d \sqrt {d+e x} \left (23 d^2+11 d e x+3 e^2 x^2\right )+\frac {2}{7} (d+e x)^{7/2}\right ) (7 A b e-4 A c d+2 b B d)-\frac {2 d \left (b c (2 B d-3 A e)-4 A c^2 d+5 b^2 B e\right ) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt {c} \sqrt {d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )}{c^{7/2} (c d-b e)}}{210 b^2}+\frac {c (d+e x)^{9/2} (A b e-2 A c d+b B d)}{b (b+c x) (b e-c d)}+\frac {A (d+e x)^{9/2}}{x (b+c x)}}{b d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x]
[Out]
-(((c*(b*B*d - 2*A*c*d + A*b*e)*(d + e*x)^(9/2))/(b*(-(c*d) + b*e)*(b + c*x)) + (A*(d + e*x)^(9/2))/(x*(b + c*
x)) - (105*(2*b*B*d - 4*A*c*d + 7*A*b*e)*((2*(d + e*x)^(7/2))/7 + (2*d*Sqrt[d + e*x]*(23*d^2 + 11*d*e*x + 3*e^
2*x^2))/15 - 2*d^(7/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - (2*d*(-4*A*c^2*d + 5*b^2*B*e + b*c*(2*B*d - 3*A*e))*(
15*c^(7/2)*(d + e*x)^(7/2) + 7*(c*d - b*e)*(3*c^(5/2)*(d + e*x)^(5/2) + 5*(c*d - b*e)*(Sqrt[c]*Sqrt[d + e*x]*(
4*c*d - 3*b*e + c*e*x) - 3*(c*d - b*e)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]))))/(c^(7/2)*(c*
d - b*e)))/(210*b^2))/(b*d))
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IntegrateAlgebraic [A] time = 0.77, size = 501, normalized size = 1.72 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-7 A b d^{5/2} e+4 A c d^{7/2}-2 b B d^{7/2}\right )}{b^3}+\frac {\left (4 A c^2 d (b e-c d)^{5/2}+3 A b c e (b e-c d)^{5/2}-5 b^2 B e (b e-c d)^{5/2}-2 b B c d (b e-c d)^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 c^{7/2}}-\frac {\sqrt {d+e x} \left (-9 A b^3 c e^3 (d+e x)+9 A b^3 c d e^3-15 A b^2 c^2 d^2 e^2-6 A b^2 c^2 e^2 (d+e x)^2+21 A b^2 c^2 d e^2 (d+e x)+12 A b c^3 d^3 e-9 A b c^3 d^2 e (d+e x)-6 A c^4 d^4+6 A c^4 d^3 (d+e x)+15 b^4 B e^3 (d+e x)-15 b^4 B d e^3+39 b^3 B c d^2 e^2+10 b^3 B c e^2 (d+e x)^2-49 b^3 B c d e^2 (d+e x)-27 b^2 B c^2 d^3 e+43 b^2 B c^2 d^2 e (d+e x)-2 b^2 B c^2 e (d+e x)^3-14 b^2 B c^2 d e (d+e x)^2+3 b B c^3 d^4-3 b B c^3 d^3 (d+e x)\right )}{3 b^2 c^3 x (b e+c (d+e x)-c d)} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x]
[Out]
-1/3*(Sqrt[d + e*x]*(3*b*B*c^3*d^4 - 6*A*c^4*d^4 - 27*b^2*B*c^2*d^3*e + 12*A*b*c^3*d^3*e + 39*b^3*B*c*d^2*e^2
- 15*A*b^2*c^2*d^2*e^2 - 15*b^4*B*d*e^3 + 9*A*b^3*c*d*e^3 - 3*b*B*c^3*d^3*(d + e*x) + 6*A*c^4*d^3*(d + e*x) +
43*b^2*B*c^2*d^2*e*(d + e*x) - 9*A*b*c^3*d^2*e*(d + e*x) - 49*b^3*B*c*d*e^2*(d + e*x) + 21*A*b^2*c^2*d*e^2*(d
+ e*x) + 15*b^4*B*e^3*(d + e*x) - 9*A*b^3*c*e^3*(d + e*x) - 14*b^2*B*c^2*d*e*(d + e*x)^2 + 10*b^3*B*c*e^2*(d +
e*x)^2 - 6*A*b^2*c^2*e^2*(d + e*x)^2 - 2*b^2*B*c^2*e*(d + e*x)^3))/(b^2*c^3*x*(-(c*d) + b*e + c*(d + e*x))) +
((-2*b*B*c*d*(-(c*d) + b*e)^(5/2) + 4*A*c^2*d*(-(c*d) + b*e)^(5/2) - 5*b^2*B*e*(-(c*d) + b*e)^(5/2) + 3*A*b*c
*e*(-(c*d) + b*e)^(5/2))*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(b^3*c^(7/2)) + ((-2*
b*B*d^(7/2) + 4*A*c*d^(7/2) - 7*A*b*d^(5/2)*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3
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fricas [A] time = 73.70, size = 2104, normalized size = 7.21
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
[-1/6*(3*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*d^2*e - 2*(4*B*b^3*c^2 - A*b^2*c^3)*d*e^2 + (5*
B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (2*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*b^
4*c - A*b^3*c^2)*d*e^2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e - 2*sqrt(e
*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 3*((7*A*b*c^4*d^2*e + 2*(B*b*c^4 - 2*A*c^5)*d^3)*x^2 + (7*A*b^2*c^
3*d^2*e + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*B*b^3*
c^2*e^3*x^3 - 3*A*b^2*c^3*d^3 + 2*(10*B*b^3*c^2*d*e^2 - (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^2*c^3 - 2
*A*b*c^4)*d^3 - 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*e + (29*B*b^4*c - 9*A*b^3*c^2)*d*e^2 - 3*(5*B*b^5 - 3*A*b^4*c)*e
^3)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x), 1/6*(6*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*
d^2*e - 2*(4*B*b^3*c^2 - A*b^2*c^3)*d*e^2 + (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (2*(B*b^2*c^3 - 2*A*b*c^4)*d^
3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*b^4*c - A*b^3*c^2)*d*e^2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(-(c
*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*((7*A*b*c^4*d^2*e + 2*(B*b*c^4 - 2*
A*c^5)*d^3)*x^2 + (7*A*b^2*c^3*d^2*e + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sq
rt(d) + 2*d)/x) + 2*(2*B*b^3*c^2*e^3*x^3 - 3*A*b^2*c^3*d^3 + 2*(10*B*b^3*c^2*d*e^2 - (5*B*b^4*c - 3*A*b^3*c^2)
*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*e + (29*B*b^4*c - 9*A*b^3*c^2)*d*e^
2 - 3*(5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x), 1/6*(6*((7*A*b*c^4*d^2*e + 2*(B*
b*c^4 - 2*A*c^5)*d^3)*x^2 + (7*A*b^2*c^3*d^2*e + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(-d)*arctan(sqrt(e*x +
d)*sqrt(-d)/d) - 3*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*d^2*e - 2*(4*B*b^3*c^2 - A*b^2*c^3)*d
*e^2 + (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (2*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e -
2*(4*B*b^4*c - A*b^3*c^2)*d*e^2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e
- 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 2*(2*B*b^3*c^2*e^3*x^3 - 3*A*b^2*c^3*d^3 + 2*(10*B*b^3*c
^2*d*e^2 - (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 9*(B*b^3*c^2 - A*b^2*c^3)*d^2
*e + (29*B*b^4*c - 9*A*b^3*c^2)*d*e^2 - 3*(5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*
x), 1/3*(3*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*d^2*e - 2*(4*B*b^3*c^2 - A*b^2*c^3)*d*e^2 + (
5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (2*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*
b^4*c - A*b^3*c^2)*d*e^2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c
*d - b*e)/c)/(c*d - b*e)) + 3*((7*A*b*c^4*d^2*e + 2*(B*b*c^4 - 2*A*c^5)*d^3)*x^2 + (7*A*b^2*c^3*d^2*e + 2*(B*b
^2*c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*B*b^3*c^2*e^3*x^3 - 3*A*b^2*c^3*d^3
+ 2*(10*B*b^3*c^2*d*e^2 - (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 9*(B*b^3*c^2
- A*b^2*c^3)*d^2*e + (29*B*b^4*c - 9*A*b^3*c^2)*d*e^2 - 3*(5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(e*x + d))/(b^3*c^
4*x^2 + b^4*c^3*x)]
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giac [B] time = 0.31, size = 639, normalized size = 2.19 \begin {gather*} \frac {{\left (2 \, B b d^{4} - 4 \, A c d^{4} + 7 \, A b d^{3} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{4} d^{4} - 4 \, A c^{5} d^{4} - B b^{2} c^{3} d^{3} e + 9 \, A b c^{4} d^{3} e - 9 \, B b^{3} c^{2} d^{2} e^{2} - 3 \, A b^{2} c^{3} d^{2} e^{2} + 13 \, B b^{4} c d e^{3} - 5 \, A b^{3} c^{2} d e^{3} - 5 \, B b^{5} e^{4} + 3 \, A b^{4} c e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{3}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B c^{4} e^{2} + 9 \, \sqrt {x e + d} B c^{4} d e^{2} - 6 \, \sqrt {x e + d} B b c^{3} e^{3} + 3 \, \sqrt {x e + d} A c^{4} e^{3}\right )}}{3 \, c^{6}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} B b c^{3} d^{3} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{4} d^{3} e - \sqrt {x e + d} B b c^{3} d^{4} e + 2 \, \sqrt {x e + d} A c^{4} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c^{2} d^{2} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{3} d^{2} e^{2} + 3 \, \sqrt {x e + d} B b^{2} c^{2} d^{3} e^{2} - 4 \, \sqrt {x e + d} A b c^{3} d^{3} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} c d e^{3} - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c^{2} d e^{3} - 3 \, \sqrt {x e + d} B b^{3} c d^{2} e^{3} + 3 \, \sqrt {x e + d} A b^{2} c^{2} d^{2} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} e^{4} + {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} c e^{4} + \sqrt {x e + d} B b^{4} d e^{4} - \sqrt {x e + d} A b^{3} c d e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )} b^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
(2*B*b*d^4 - 4*A*c*d^4 + 7*A*b*d^3*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) - (2*B*b*c^4*d^4 - 4*A*c^5
*d^4 - B*b^2*c^3*d^3*e + 9*A*b*c^4*d^3*e - 9*B*b^3*c^2*d^2*e^2 - 3*A*b^2*c^3*d^2*e^2 + 13*B*b^4*c*d*e^3 - 5*A*
b^3*c^2*d*e^3 - 5*B*b^5*e^4 + 3*A*b^4*c*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e
)*b^3*c^3) + 2/3*((x*e + d)^(3/2)*B*c^4*e^2 + 9*sqrt(x*e + d)*B*c^4*d*e^2 - 6*sqrt(x*e + d)*B*b*c^3*e^3 + 3*sq
rt(x*e + d)*A*c^4*e^3)/c^6 + ((x*e + d)^(3/2)*B*b*c^3*d^3*e - 2*(x*e + d)^(3/2)*A*c^4*d^3*e - sqrt(x*e + d)*B*
b*c^3*d^4*e + 2*sqrt(x*e + d)*A*c^4*d^4*e - 3*(x*e + d)^(3/2)*B*b^2*c^2*d^2*e^2 + 3*(x*e + d)^(3/2)*A*b*c^3*d^
2*e^2 + 3*sqrt(x*e + d)*B*b^2*c^2*d^3*e^2 - 4*sqrt(x*e + d)*A*b*c^3*d^3*e^2 + 3*(x*e + d)^(3/2)*B*b^3*c*d*e^3
- 3*(x*e + d)^(3/2)*A*b^2*c^2*d*e^3 - 3*sqrt(x*e + d)*B*b^3*c*d^2*e^3 + 3*sqrt(x*e + d)*A*b^2*c^2*d^2*e^3 - (x
*e + d)^(3/2)*B*b^4*e^4 + (x*e + d)^(3/2)*A*b^3*c*e^4 + sqrt(x*e + d)*B*b^4*d*e^4 - sqrt(x*e + d)*A*b^3*c*d*e^
4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c^3)
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maple [B] time = 0.09, size = 823, normalized size = 2.82 \begin {gather*} -\frac {3 A b \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c^{2}}+\frac {3 A \,d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b}-\frac {9 A c \,d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {4 A \,c^{2} d^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {5 A d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c}+\frac {5 B \,b^{2} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c^{3}}-\frac {13 B b d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c^{2}}+\frac {B \,d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b}-\frac {2 B c \,d^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {9 B \,d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c}+\frac {\sqrt {e x +d}\, A b \,e^{4}}{\left (c e x +b e \right ) c^{2}}+\frac {3 \sqrt {e x +d}\, A \,d^{2} e^{2}}{\left (c e x +b e \right ) b}-\frac {\sqrt {e x +d}\, A c \,d^{3} e}{\left (c e x +b e \right ) b^{2}}-\frac {3 \sqrt {e x +d}\, A d \,e^{3}}{\left (c e x +b e \right ) c}-\frac {\sqrt {e x +d}\, B \,b^{2} e^{4}}{\left (c e x +b e \right ) c^{3}}+\frac {3 \sqrt {e x +d}\, B b d \,e^{3}}{\left (c e x +b e \right ) c^{2}}+\frac {\sqrt {e x +d}\, B \,d^{3} e}{\left (c e x +b e \right ) b}-\frac {3 \sqrt {e x +d}\, B \,d^{2} e^{2}}{\left (c e x +b e \right ) c}-\frac {7 A \,d^{\frac {5}{2}} e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 A c \,d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3}}-\frac {2 B \,d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {2 \sqrt {e x +d}\, A \,e^{3}}{c^{2}}-\frac {4 \sqrt {e x +d}\, B b \,e^{3}}{c^{3}}+\frac {6 \sqrt {e x +d}\, B d \,e^{2}}{c^{2}}-\frac {\sqrt {e x +d}\, A \,d^{3}}{b^{2} x}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,e^{2}}{3 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^2,x)
[Out]
2*e^3/c^2*A*(e*x+d)^(1/2)+2/3*e^2/c^2*B*(e*x+d)^(3/2)-2*d^(7/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*B+e/b/((b*e
-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^3-3*e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d+4/b^3*c
^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^4-2/b^2*c/((b*e-c*d)*c)^(1/2)*arctan((e
*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^4+e^4*b/c^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A-e^4*b^2/c^3*(e*x+d)^(1/2)/(c*e*
x+b*e)*B-3*e^4*b/c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A+5*e^4*b^2/c^3/((b*e-c*d
)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B-d^3/b^2*A*(e*x+d)^(1/2)/x+4*d^(7/2)/b^3*arctanh((e*x+
d)^(1/2)/d^(1/2))*A*c-7*e*d^(5/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*A-4*e^3/c^3*B*b*(e*x+d)^(1/2)+6*e^2/c^2*B
*d*(e*x+d)^(1/2)+3*e^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^2-3*e^2/c*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^2+e/b*(e*x+d)^(
1/2)/(c*e*x+b*e)*B*d^3+5*e^3/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d+3*e^2/b/((b
*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2+9*e^2/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^
(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^2+3*e^3*b/c^2*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d-9*e/b^2*c/((b*e-c*d)*c)^(1/2)*arc
tan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^3-13*e^3*b/c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d
)*c)^(1/2)*c)*B*d-e/b^2*c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^3
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?
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mupad [B] time = 3.29, size = 7328, normalized size = 25.10
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x)
[Out]
(((d + e*x)^(3/2)*(B*b^4*e^4 - A*b^3*c*e^4 + 2*A*c^4*d^3*e - 3*A*b*c^3*d^2*e^2 + 3*A*b^2*c^2*d*e^3 + 3*B*b^2*c
^2*d^2*e^2 - B*b*c^3*d^3*e - 3*B*b^3*c*d*e^3))/b^2 - ((d + e*x)^(1/2)*(2*A*c^4*d^4*e + B*b^4*d*e^4 - 4*A*b*c^3
*d^3*e^2 - 3*B*b^3*c*d^2*e^3 + 3*A*b^2*c^2*d^2*e^3 + 3*B*b^2*c^2*d^3*e^2 - A*b^3*c*d*e^4 - B*b*c^3*d^4*e))/b^2
)/((2*c^4*d - b*c^3*e)*(d + e*x) - c^4*(d + e*x)^2 - c^4*d^2 + b*c^3*d*e) + ((2*A*e^3 - 2*B*d*e^2)/c^2 + (2*B*
e^2*(4*c^2*d - 2*b*c*e))/c^4)*(d + e*x)^(1/2) + (atan(((((((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6
*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B
*b^9*c^5*d^2*e^5)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*
d + 2*B*b*d))/(b^7*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*B^2*b^10*
e^10 + 9*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b
^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^
3 - 35*B^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B
^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 -
32*A*B*b*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c
^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5))*(d^5)^(1/
2)*(7*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^3) - (((((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4
*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5
*d^2*e^5)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*
b*d))/(b^7*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3) - (2*(d + e*x)^(1/2)*(25*B^2*b^10*e^10 + 9
*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d
^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B
^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c
^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b
*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c^6*d^5*e
^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5))*(d^5)^(1/2)*(7*A*
b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^3))/((2*(32*A^3*c^10*d^11*e^3 + 50*B^3*b^10*d^4*e^10 + 262*A^3*b^2*c^8*d^9*e
^5 + 141*A^3*b^3*c^7*d^8*e^6 - 658*A^3*b^4*c^6*d^7*e^7 + 413*A^3*b^5*c^5*d^6*e^8 + 169*A^3*b^6*c^4*d^5*e^9 - 2
46*A^3*b^7*c^3*d^4*e^10 + 63*A^3*b^8*c^2*d^3*e^11 - 4*B^3*b^3*c^7*d^11*e^3 - 34*B^3*b^4*c^6*d^10*e^4 + 88*B^3*
b^5*c^5*d^9*e^5 + 90*B^3*b^6*c^4*d^8*e^6 - 448*B^3*b^7*c^3*d^7*e^7 + 518*B^3*b^8*c^2*d^6*e^8 + 175*A*B^2*b^10*
d^3*e^11 - 176*A^3*b*c^9*d^10*e^4 - 260*B^3*b^9*c*d^5*e^9 + 24*A*B^2*b^2*c^8*d^11*e^3 + 92*A*B^2*b^3*c^7*d^10*
e^4 - 605*A*B^2*b^4*c^6*d^9*e^5 + 594*A*B^2*b^5*c^5*d^8*e^6 + 1113*A*B^2*b^6*c^4*d^7*e^7 - 2912*A*B^2*b^7*c^3*
d^6*e^8 + 2589*A*B^2*b^8*c^2*d^5*e^9 + 40*A^2*B*b^2*c^8*d^10*e^4 + 727*A^2*B*b^3*c^7*d^9*e^5 - 2133*A^2*B*b^4*
c^6*d^8*e^6 + 1953*A^2*B*b^5*c^5*d^7*e^7 + 287*A^2*B*b^6*c^4*d^6*e^8 - 1650*A^2*B*b^7*c^3*d^5*e^9 + 1034*A^2*B
*b^8*c^2*d^4*e^10 - 1070*A*B^2*b^9*c*d^4*e^10 - 48*A^2*B*b*c^9*d^11*e^3 - 210*A^2*B*b^9*c*d^3*e^11))/(b^6*c^5)
+ (((((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d
^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5*d^2*e^5)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^
6*c^8*d*e^2)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*
c*d + 2*B*b*d))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*B^2*b^10*e^10 + 9*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 15
4*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*
c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5
+ 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b
*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*
B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^
4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3) + (((((12*
A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*
B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5*d^2*e^5)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2
)*(d^5)^(1/2)*(d + e*x)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b
*d))/(2*b^3) - (2*(d + e*x)^(1/2)*(25*B^2*b^10*e^10 + 9*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c
^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*c^4*d^2*e^8
+ 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^
6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b*c^9*d^7*e^
3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d
^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^4*d^3*e^7 -
154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5))*(d^5)^(1/2)*(7*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3)))*(d^5)^(1/2)*(7*A*b*
e - 4*A*c*d + 2*B*b*d)*1i)/b^3 + (2*B*e^2*(d + e*x)^(3/2))/(3*c^2) + (atan((((-c^7*(b*e - c*d)^5)^(1/2)*((2*(d
+ e*x)^(1/2)*(25*B^2*b^10*e^10 + 9*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2*
b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8
*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*
B^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d
*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*
c^7*d^6*e^4 - 280*A*B*b^4*c^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^4*d^3*e^7 - 154*A*B*b^7*c^3*d^2
*e^8))/(b^4*c^5) + ((-c^7*(b*e - c*d)^5)^(1/2)*((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^
3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5*d^
2*e^5)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d -
5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d))/(b^7*c^12))*(4*A*c^2*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d))/(2*b^3*c^7))*
(4*A*c^2*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d)*1i)/(2*b^3*c^7) + ((-c^7*(b*e - c*d)^5)^(1/2)*((2*(d + e*x)^(1
/2)*(25*B^2*b^10*e^10 + 9*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^
5*e^5 - 105*A^2*b^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 + 7*A^2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B
^2*b^3*c^7*d^7*e^3 - 35*B^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^
3*d^3*e^7 + 259*B^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 - 128*A^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*
A*B*b^9*c*e^10 - 32*A*B*b*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^
4 - 280*A*B*b^4*c^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*A*B*b^6*c^4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^
4*c^5) - ((-c^7*(b*e - c*d)^5)^(1/2)*((12*A*b^9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^3 + 16*A*b
^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*B*b^7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5*d^2*e^5)/(b^
6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d - 5*B*b^2*e
+ 3*A*b*c*e - 2*B*b*c*d))/(b^7*c^12))*(4*A*c^2*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d))/(2*b^3*c^7))*(4*A*c^2*d
- 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d)*1i)/(2*b^3*c^7))/((2*(32*A^3*c^10*d^11*e^3 + 50*B^3*b^10*d^4*e^10 + 262*
A^3*b^2*c^8*d^9*e^5 + 141*A^3*b^3*c^7*d^8*e^6 - 658*A^3*b^4*c^6*d^7*e^7 + 413*A^3*b^5*c^5*d^6*e^8 + 169*A^3*b^
6*c^4*d^5*e^9 - 246*A^3*b^7*c^3*d^4*e^10 + 63*A^3*b^8*c^2*d^3*e^11 - 4*B^3*b^3*c^7*d^11*e^3 - 34*B^3*b^4*c^6*d
^10*e^4 + 88*B^3*b^5*c^5*d^9*e^5 + 90*B^3*b^6*c^4*d^8*e^6 - 448*B^3*b^7*c^3*d^7*e^7 + 518*B^3*b^8*c^2*d^6*e^8
+ 175*A*B^2*b^10*d^3*e^11 - 176*A^3*b*c^9*d^10*e^4 - 260*B^3*b^9*c*d^5*e^9 + 24*A*B^2*b^2*c^8*d^11*e^3 + 92*A*
B^2*b^3*c^7*d^10*e^4 - 605*A*B^2*b^4*c^6*d^9*e^5 + 594*A*B^2*b^5*c^5*d^8*e^6 + 1113*A*B^2*b^6*c^4*d^7*e^7 - 29
12*A*B^2*b^7*c^3*d^6*e^8 + 2589*A*B^2*b^8*c^2*d^5*e^9 + 40*A^2*B*b^2*c^8*d^10*e^4 + 727*A^2*B*b^3*c^7*d^9*e^5
- 2133*A^2*B*b^4*c^6*d^8*e^6 + 1953*A^2*B*b^5*c^5*d^7*e^7 + 287*A^2*B*b^6*c^4*d^6*e^8 - 1650*A^2*B*b^7*c^3*d^5
*e^9 + 1034*A^2*B*b^8*c^2*d^4*e^10 - 1070*A*B^2*b^9*c*d^4*e^10 - 48*A^2*B*b*c^9*d^11*e^3 - 210*A^2*B*b^9*c*d^3
*e^11))/(b^6*c^5) + ((-c^7*(b*e - c*d)^5)^(1/2)*((2*(d + e*x)^(1/2)*(25*B^2*b^10*e^10 + 9*A^2*b^8*c^2*e^10 + 3
2*A^2*c^10*d^8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d^4*e^6 + 84*A^2*b^5*c
^5*d^3*e^7 + 7*A^2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B^2*b^4*c^6*d^6*e^4 +
70*B^2*b^5*c^5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c^2*d^2*e^8 - 130*B^2*
b^9*c*d*e^9 - 128*A^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b*c^9*d^8*e^2 + 128*A*
B*b^8*c^2*d*e^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c^6*d^5*e^5 + 350*A*B*b^5*c^5*
d^4*e^6 - 84*A*B*b^6*c^4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5) + ((-c^7*(b*e - c*d)^5)^(1/2)*((12*A*b^
9*c^5*d*e^6 - 20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*B*b^
7*c^7*d^4*e^3 - 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5*d^2*e^5)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-
c^7*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d))/(b^7*c^12))*(4*A*c^2
*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d))/(2*b^3*c^7))*(4*A*c^2*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d))/(2*b^3*
c^7) - ((-c^7*(b*e - c*d)^5)^(1/2)*((2*(d + e*x)^(1/2)*(25*B^2*b^10*e^10 + 9*A^2*b^8*c^2*e^10 + 32*A^2*c^10*d^
8*e^2 + 154*A^2*b^2*c^8*d^6*e^4 - 14*A^2*b^3*c^7*d^5*e^5 - 105*A^2*b^4*c^6*d^4*e^6 + 84*A^2*b^5*c^5*d^3*e^7 +
7*A^2*b^6*c^4*d^2*e^8 + 8*B^2*b^2*c^8*d^8*e^2 - 4*B^2*b^3*c^7*d^7*e^3 - 35*B^2*b^4*c^6*d^6*e^4 + 70*B^2*b^5*c^
5*d^5*e^5 + 35*B^2*b^6*c^4*d^4*e^6 - 224*B^2*b^7*c^3*d^3*e^7 + 259*B^2*b^8*c^2*d^2*e^8 - 130*B^2*b^9*c*d*e^9 -
128*A^2*b*c^9*d^7*e^3 - 30*A^2*b^7*c^3*d*e^9 - 30*A*B*b^9*c*e^10 - 32*A*B*b*c^9*d^8*e^2 + 128*A*B*b^8*c^2*d*e
^9 + 72*A*B*b^2*c^8*d^7*e^3 + 42*A*B*b^3*c^7*d^6*e^4 - 280*A*B*b^4*c^6*d^5*e^5 + 350*A*B*b^5*c^5*d^4*e^6 - 84*
A*B*b^6*c^4*d^3*e^7 - 154*A*B*b^7*c^3*d^2*e^8))/(b^4*c^5) - ((-c^7*(b*e - c*d)^5)^(1/2)*((12*A*b^9*c^5*d*e^6 -
20*B*b^10*c^4*d*e^6 - 8*A*b^6*c^8*d^4*e^3 + 16*A*b^7*c^7*d^3*e^4 - 20*A*b^8*c^6*d^2*e^5 + 4*B*b^7*c^7*d^4*e^3
- 36*B*b^8*c^6*d^3*e^4 + 52*B*b^9*c^5*d^2*e^5)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e - c*
d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d))/(b^7*c^12))*(4*A*c^2*d - 5*B*b^2*
e + 3*A*b*c*e - 2*B*b*c*d))/(2*b^3*c^7))*(4*A*c^2*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d))/(2*b^3*c^7)))*(-c^7*
(b*e - c*d)^5)^(1/2)*(4*A*c^2*d - 5*B*b^2*e + 3*A*b*c*e - 2*B*b*c*d)*1i)/(b^3*c^7)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**(7/2)/(c*x**2+b*x)**2,x)
[Out]
Timed out
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