3.13.38 \(\int \frac {(A+B x) (d+e x)^{9/2}}{(b x+c x^2)^2} \, dx\) [1238]

Optimal. Leaf size=386 \[ \frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac {d^{7/2} (2 b B d-4 A c d+9 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{7/2} \left (4 A c^2 d-7 b^2 B e-b c (2 B d-5 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{9/2}} \]

[Out]

1/3*e*(6*A*c^3*d^2-7*b^3*B*e^2-3*b*c^2*d*(2*A*e+B*d)+b^2*c*e*(5*A*e+12*B*d))*(e*x+d)^(3/2)/b^2/c^3+1/5*e*(10*A
*c^2*d+7*b^2*B*e-5*b*c*(A*e+B*d))*(e*x+d)^(5/2)/b^2/c^2-(e*x+d)^(7/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d
))*x)/b^2/c/(c*x^2+b*x)-d^(7/2)*(9*A*b*e-4*A*c*d+2*B*b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3-(-b*e+c*d)^(7/2)*
(4*A*c^2*d-7*b^2*B*e-b*c*(-5*A*e+2*B*d))*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/c^(9/2)+e*(2*A*c^
4*d^3+7*b^4*B*e^3-b*c^3*d^2*(3*A*e+B*d)-b^3*c*e^2*(5*A*e+19*B*d)+b^2*c^2*d*e*(11*A*e+15*B*d))*(e*x+d)^(1/2)/b^
2/c^4

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Rubi [A]
time = 0.79, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {832, 838, 840, 1180, 214} \begin {gather*} -\frac {d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (9 A b e-4 A c d+2 b B d)}{b^3}-\frac {(d+e x)^{7/2} \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)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 b^2 c^2}-\frac {(c d-b e)^{7/2} \left (-b c (2 B d-5 A e)+4 A c^2 d-7 b^2 B e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{9/2}}+\frac {e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 b^2 c^3}+\frac {e \sqrt {d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{b^2 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x]

[Out]

(e*(2*A*c^4*d^3 + 7*b^4*B*e^3 - b*c^3*d^2*(B*d + 3*A*e) - b^3*c*e^2*(19*B*d + 5*A*e) + b^2*c^2*d*e*(15*B*d + 1
1*A*e))*Sqrt[d + e*x])/(b^2*c^4) + (e*(6*A*c^3*d^2 - 7*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(12*B*d +
5*A*e))*(d + e*x)^(3/2))/(3*b^2*c^3) + (e*(10*A*c^2*d + 7*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(5/2))/(5*b^
2*c^2) - ((d + e*x)^(7/2)*(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^(7
/2)*(2*b*B*d - 4*A*c*d + 9*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^(7/2)*(4*A*c^2*d - 7*b^2*
B*e - b*c*(2*B*d - 5*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(9/2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(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] &&
NeQ[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 838

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 840

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 1180

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)^{9/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{7/2} \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)^{5/2} \left (\frac {1}{2} c d (2 b B d-4 A c d+9 A b e)+\frac {1}{2} e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \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^2 d^2 (2 b B d-4 A c d+9 A b e)+\frac {1}{2} e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \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^3 d^3 (2 b B d-4 A c d+9 A b e)+\frac {1}{2} e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^3}\\ &=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \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^4 d^4 (2 b B d-4 A c d+9 A b e)-\frac {1}{2} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^4}\\ &=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \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 \text {Subst}\left (\int \frac {\frac {1}{2} c^4 d^4 e (2 b B d-4 A c d+9 A b e)+\frac {1}{2} d e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )-\frac {1}{2} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 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^4}\\ &=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \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^4 (2 b B d-4 A c d+9 A b e)\right ) \text {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 (2 \left (\frac {1}{4} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )+\frac {\frac {1}{2} e (-2 c d+b e) \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )+2 c \left (\frac {1}{2} c^4 d^4 e (2 b B d-4 A c d+9 A b e)+\frac {1}{2} d e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )\right )}{2 b e}\right )\right ) \text {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^2 c^4}\\ &=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \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^{7/2} (2 b B d-4 A c d+9 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {(c d-b e)^{7/2} \left (2 b B c d-4 A c^2 d+7 b^2 B e-5 A b c e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.92, size = 336, normalized size = 0.87 \begin {gather*} \frac {\frac {b \sqrt {d+e x} \left (-5 A c \left (6 c^4 d^4 x+15 b^4 e^4 x+3 b c^3 d^3 (d-4 e x)+2 b^3 c e^3 x (-19 d+5 e x)-2 b^2 c^2 e^2 x \left (-9 d^2+13 d e x+e^2 x^2\right )\right )+b B x \left (15 c^4 d^4+105 b^4 e^4+10 b^3 c e^3 (-32 d+7 e x)-2 b^2 c^2 e^2 \left (-153 d^2+109 d e x+7 e^2 x^2\right )+6 b c^3 e \left (-10 d^3+36 d^2 e x+7 d e^2 x^2+e^3 x^3\right )\right )\right )}{c^4 x (b+c x)}+\frac {15 (-c d+b e)^{7/2} \left (-2 b B c d+4 A c^2 d-7 b^2 B e+5 A b c e\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{9/2}}-15 d^{7/2} (2 b B d-4 A c d+9 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{15 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x]

[Out]

((b*Sqrt[d + e*x]*(-5*A*c*(6*c^4*d^4*x + 15*b^4*e^4*x + 3*b*c^3*d^3*(d - 4*e*x) + 2*b^3*c*e^3*x*(-19*d + 5*e*x
) - 2*b^2*c^2*e^2*x*(-9*d^2 + 13*d*e*x + e^2*x^2)) + b*B*x*(15*c^4*d^4 + 105*b^4*e^4 + 10*b^3*c*e^3*(-32*d + 7
*e*x) - 2*b^2*c^2*e^2*(-153*d^2 + 109*d*e*x + 7*e^2*x^2) + 6*b*c^3*e*(-10*d^3 + 36*d^2*e*x + 7*d*e^2*x^2 + e^3
*x^3))))/(c^4*x*(b + c*x)) + (15*(-(c*d) + b*e)^(7/2)*(-2*b*B*c*d + 4*A*c^2*d - 7*b^2*B*e + 5*A*b*c*e)*ArcTan[
(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(9/2) - 15*d^(7/2)*(2*b*B*d - 4*A*c*d + 9*A*b*e)*ArcTanh[Sqrt[d
+ e*x]/Sqrt[d]])/(15*b^3)

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Maple [A]
time = 0.68, size = 548, normalized size = 1.42 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)

[Out]

2*e^2*(-1/c^4*(-1/5*B*c^2*(e*x+d)^(5/2)-1/3*A*c^2*e*(e*x+d)^(3/2)+2/3*B*b*c*e*(e*x+d)^(3/2)-B*c^2*d*(e*x+d)^(3
/2)+2*A*b*c*e^2*(e*x+d)^(1/2)-4*A*c^2*d*e*(e*x+d)^(1/2)-3*B*b^2*e^2*(e*x+d)^(1/2)+8*B*b*c*d*e*(e*x+d)^(1/2)-6*
B*c^2*d^2*(e*x+d)^(1/2))-d^4/e^2/b^3*(1/2*A*b*(e*x+d)^(1/2)/x+1/2*(9*A*b*e-4*A*c*d+2*B*b*d)/d^(1/2)*arctanh((e
*x+d)^(1/2)/d^(1/2)))+1/c^4/e^2/b^3*((-1/2*A*b^5*c*e^5+2*A*b^4*c^2*d*e^4-3*A*b^3*c^3*d^2*e^3+2*A*b^2*c^4*d^3*e
^2-1/2*A*b*c^5*d^4*e+1/2*b^6*B*e^5-2*b^5*B*c*d*e^4+3*B*b^4*c^2*d^2*e^3-2*B*b^3*c^3*d^3*e^2+1/2*B*b^2*c^4*d^4*e
)*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(5*A*b^5*c*e^5-16*A*b^4*c^2*d*e^4+14*A*b^3*c^3*d^2*e^3+4*A*b^2*c^4*d^3
*e^2-11*A*b*c^5*d^4*e+4*A*c^6*d^5-7*B*b^6*e^5+26*B*b^5*c*d*e^4-34*B*b^4*c^2*d^2*e^3+16*B*b^3*c^3*d^3*e^2+B*b^2
*c^4*d^4*e-2*B*b*c^5*d^5)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))))

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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)^(9/2)/(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(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [F(-1)] Timed out
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)^(9/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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)**(9/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (380) = 760\).
time = 0.83, size = 846, normalized size = 2.19 \begin {gather*} \frac {{\left (2 \, B b d^{5} - 4 \, A c d^{5} + 9 \, A b d^{4} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{5} d^{5} - 4 \, A c^{6} d^{5} - B b^{2} c^{4} d^{4} e + 11 \, A b c^{5} d^{4} e - 16 \, B b^{3} c^{3} d^{3} e^{2} - 4 \, A b^{2} c^{4} d^{3} e^{2} + 34 \, B b^{4} c^{2} d^{2} e^{3} - 14 \, A b^{3} c^{3} d^{2} e^{3} - 26 \, B b^{5} c d e^{4} + 16 \, A b^{4} c^{2} d e^{4} + 7 \, B b^{6} e^{5} - 5 \, A b^{5} c e^{5}\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^{4}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} B b c^{4} d^{4} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{5} d^{4} e - \sqrt {x e + d} B b c^{4} d^{5} e + 2 \, \sqrt {x e + d} A c^{5} d^{5} e - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c^{3} d^{3} e^{2} + 4 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{4} d^{3} e^{2} + 4 \, \sqrt {x e + d} B b^{2} c^{3} d^{4} e^{2} - 5 \, \sqrt {x e + d} A b c^{4} d^{4} e^{2} + 6 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} c^{2} d^{2} e^{3} - 6 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c^{3} d^{2} e^{3} - 6 \, \sqrt {x e + d} B b^{3} c^{2} d^{3} e^{3} + 6 \, \sqrt {x e + d} A b^{2} c^{3} d^{3} e^{3} - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} c d e^{4} + 4 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} c^{2} d e^{4} + 4 \, \sqrt {x e + d} B b^{4} c d^{2} e^{4} - 4 \, \sqrt {x e + d} A b^{3} c^{2} d^{2} e^{4} + {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} e^{5} - {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} c e^{5} - \sqrt {x e + d} B b^{5} d e^{5} + \sqrt {x e + d} A b^{4} c d e^{5}}{{\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^{4}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{8} e^{2} + 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{8} d e^{2} + 90 \, \sqrt {x e + d} B c^{8} d^{2} e^{2} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c^{7} e^{3} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{8} e^{3} - 120 \, \sqrt {x e + d} B b c^{7} d e^{3} + 60 \, \sqrt {x e + d} A c^{8} d e^{3} + 45 \, \sqrt {x e + d} B b^{2} c^{6} e^{4} - 30 \, \sqrt {x e + d} A b c^{7} e^{4}\right )}}{15 \, c^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

(2*B*b*d^5 - 4*A*c*d^5 + 9*A*b*d^4*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) - (2*B*b*c^5*d^5 - 4*A*c^6
*d^5 - B*b^2*c^4*d^4*e + 11*A*b*c^5*d^4*e - 16*B*b^3*c^3*d^3*e^2 - 4*A*b^2*c^4*d^3*e^2 + 34*B*b^4*c^2*d^2*e^3
- 14*A*b^3*c^3*d^2*e^3 - 26*B*b^5*c*d*e^4 + 16*A*b^4*c^2*d*e^4 + 7*B*b^6*e^5 - 5*A*b^5*c*e^5)*arctan(sqrt(x*e
+ d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3*c^4) + ((x*e + d)^(3/2)*B*b*c^4*d^4*e - 2*(x*e + d)^(3/
2)*A*c^5*d^4*e - sqrt(x*e + d)*B*b*c^4*d^5*e + 2*sqrt(x*e + d)*A*c^5*d^5*e - 4*(x*e + d)^(3/2)*B*b^2*c^3*d^3*e
^2 + 4*(x*e + d)^(3/2)*A*b*c^4*d^3*e^2 + 4*sqrt(x*e + d)*B*b^2*c^3*d^4*e^2 - 5*sqrt(x*e + d)*A*b*c^4*d^4*e^2 +
6*(x*e + d)^(3/2)*B*b^3*c^2*d^2*e^3 - 6*(x*e + d)^(3/2)*A*b^2*c^3*d^2*e^3 - 6*sqrt(x*e + d)*B*b^3*c^2*d^3*e^3
+ 6*sqrt(x*e + d)*A*b^2*c^3*d^3*e^3 - 4*(x*e + d)^(3/2)*B*b^4*c*d*e^4 + 4*(x*e + d)^(3/2)*A*b^3*c^2*d*e^4 + 4
*sqrt(x*e + d)*B*b^4*c*d^2*e^4 - 4*sqrt(x*e + d)*A*b^3*c^2*d^2*e^4 + (x*e + d)^(3/2)*B*b^5*e^5 - (x*e + d)^(3/
2)*A*b^4*c*e^5 - sqrt(x*e + d)*B*b^5*d*e^5 + sqrt(x*e + d)*A*b^4*c*d*e^5)/(((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^4) + 2/15*(3*(x*e + d)^(5/2)*B*c^8*e^2 + 15*(x*e + d)^(3/2)*B*c^8*d*e^2 +
90*sqrt(x*e + d)*B*c^8*d^2*e^2 - 10*(x*e + d)^(3/2)*B*b*c^7*e^3 + 5*(x*e + d)^(3/2)*A*c^8*e^3 - 120*sqrt(x*e
+ d)*B*b*c^7*d*e^3 + 60*sqrt(x*e + d)*A*c^8*d*e^3 + 45*sqrt(x*e + d)*B*b^2*c^6*e^4 - 30*sqrt(x*e + d)*A*b*c^7*
e^4)/c^10

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Mupad [B]
time = 6.71, size = 2500, normalized size = 6.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x)

[Out]

atan(((((20*A*b^10*c^6*d*e^7 - 28*B*b^11*c^5*d*e^7 + 8*A*b^6*c^10*d^5*e^3 - 20*A*b^7*c^9*d^4*e^4 + 56*A*b^8*c^
8*d^3*e^5 - 64*A*b^9*c^7*d^2*e^6 - 4*B*b^7*c^9*d^5*e^3 + 64*B*b^8*c^8*d^4*e^4 - 136*B*b^9*c^7*d^3*e^5 + 104*B*
b^10*c^6*d^2*e^6)/(b^6*c^7) - (2*(4*b^7*c^9*e^3 - 8*b^6*c^10*d*e^2)*(d + e*x)^(1/2)*((16*A^2*c^11*d^9 - 49*B^2
*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^
2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c
^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^
3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*
b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*
A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^
8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2))/(b^4*c^7))*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2
*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^
4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 +
189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A
^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*
B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*
d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2) - (2
*(d + e*x)^(1/2)*(49*B^2*b^12*e^12 + 25*A^2*b^10*c^2*e^12 + 32*A^2*c^12*d^10*e^2 + 234*A^2*b^2*c^10*d^8*e^4 +
24*A^2*b^3*c^9*d^7*e^5 - 420*A^2*b^4*c^8*d^6*e^6 + 504*A^2*b^5*c^7*d^5*e^7 - 42*A^2*b^6*c^6*d^4*e^8 - 408*A^2*
b^7*c^5*d^3*e^9 + 396*A^2*b^8*c^4*d^2*e^10 + 8*B^2*b^2*c^10*d^10*e^2 - 4*B^2*b^3*c^9*d^9*e^3 - 63*B^2*b^4*c^8*
d^8*e^4 + 168*B^2*b^5*c^7*d^7*e^5 + 84*B^2*b^6*c^6*d^6*e^6 - 1008*B^2*b^7*c^5*d^5*e^7 + 1974*B^2*b^8*c^4*d^4*e
^8 - 1992*B^2*b^9*c^3*d^3*e^9 + 1152*B^2*b^10*c^2*d^2*e^10 - 364*B^2*b^11*c*d*e^11 - 160*A^2*b*c^11*d^9*e^3 -
160*A^2*b^9*c^3*d*e^11 - 70*A*B*b^11*c*e^12 - 32*A*B*b*c^11*d^10*e^2 + 484*A*B*b^10*c^2*d*e^11 + 88*A*B*b^2*c^
10*d^9*e^3 + 90*A*B*b^3*c^9*d^8*e^4 - 672*A*B*b^4*c^8*d^7*e^5 + 1176*A*B*b^5*c^7*d^6*e^6 - 504*A*B*b^6*c^6*d^5
*e^7 - 1092*A*B*b^7*c^5*d^4*e^8 + 1920*A*B*b^8*c^4*d^3*e^9 - 1368*A*B*b^9*c^3*d^2*e^10))/(b^4*c^7))*((16*A^2*c
^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*
d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^
7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 115
5*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3
*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^
8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*
e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2)*1i - (((20*A*b^10*c^6*d*e^7 - 28*B*b^11*c^5*d*e^7 + 8*A*b^6*
c^10*d^5*e^3 - 20*A*b^7*c^9*d^4*e^4 + 56*A*b^8*c^8*d^3*e^5 - 64*A*b^9*c^7*d^2*e^6 - 4*B*b^7*c^9*d^5*e^3 + 64*B
*b^8*c^8*d^4*e^4 - 136*B*b^9*c^7*d^3*e^5 + 104*B*b^10*c^6*d^2*e^6)/(b^6*c^7) + (2*(4*b^7*c^9*e^3 - 8*b^6*c^10*
d*e^2)*(d + e*x)^(1/2)*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b
^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5
*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^
4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B
^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A
*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c
^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2))/(b^4*c^7))*((16*A^2*c^11*d
^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e
^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 6
3*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2
*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^
8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7
*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^...

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