3.1229 \(\int \frac {(A+B x) (d+e x)^{7/2}}{b x+c x^2} \, dx\)

Optimal. Leaf size=228 \[ \frac {2 \sqrt {d+e x} \left (A c e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )+B (c d-b e)^3\right )}{c^4}-\frac {2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b c^{9/2}}+\frac {2 (d+e x)^{3/2} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{3 c^3}+\frac {2 (d+e x)^{5/2} (A c e-b B e+B c d)}{5 c^2}-\frac {2 A d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b}+\frac {2 B (d+e x)^{7/2}}{7 c} \]

[Out]

2/3*(B*(-b*e+c*d)^2+A*c*e*(-b*e+2*c*d))*(e*x+d)^(3/2)/c^3+2/5*(A*c*e-B*b*e+B*c*d)*(e*x+d)^(5/2)/c^2+2/7*B*(e*x
+d)^(7/2)/c-2*A*d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b-2*(-A*c+B*b)*(-b*e+c*d)^(7/2)*arctanh(c^(1/2)*(e*x+d)
^(1/2)/(-b*e+c*d)^(1/2))/b/c^(9/2)+2*(B*(-b*e+c*d)^3+A*c*e*(b^2*e^2-3*b*c*d*e+3*c^2*d^2))*(e*x+d)^(1/2)/c^4

________________________________________________________________________________________

Rubi [A]  time = 0.56, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {824, 826, 1166, 208} \[ \frac {2 \sqrt {d+e x} \left (A c e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )+B (c d-b e)^3\right )}{c^4}+\frac {2 (d+e x)^{5/2} (A c e-b B e+B c d)}{5 c^2}+\frac {2 (d+e x)^{3/2} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{3 c^3}-\frac {2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b c^{9/2}}-\frac {2 A d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b}+\frac {2 B (d+e x)^{7/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2),x]

[Out]

(2*(B*(c*d - b*e)^3 + A*c*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2))*Sqrt[d + e*x])/c^4 + (2*(B*(c*d - b*e)^2 + A*c*
e*(2*c*d - b*e))*(d + e*x)^(3/2))/(3*c^3) + (2*(B*c*d - b*B*e + A*c*e)*(d + e*x)^(5/2))/(5*c^2) + (2*B*(d + e*
x)^(7/2))/(7*c) - (2*A*d^(7/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b - (2*(b*B - A*c)*(c*d - b*e)^(7/2)*ArcTanh[(S
qrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*c^(9/2))

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 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}}{b x+c x^2} \, dx &=\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\int \frac {(d+e x)^{5/2} (A c d+(B c d-b B e+A c e) x)}{b x+c x^2} \, dx}{c}\\ &=\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\int \frac {(d+e x)^{3/2} \left (A c^2 d^2+\left (B (c d-b e)^2+A c e (2 c d-b e)\right ) x\right )}{b x+c x^2} \, dx}{c^2}\\ &=\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\int \frac {\sqrt {d+e x} \left (A c^3 d^3+\left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) x\right )}{b x+c x^2} \, dx}{c^3}\\ &=\frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\int \frac {A c^4 d^4+\left (B (c d-b e)^4+A c e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{c^4}\\ &=\frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {2 \operatorname {Subst}\left (\int \frac {A c^4 d^4 e-d \left (B (c d-b e)^4+A c e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )+\left (B (c d-b e)^4+A c e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\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 )}{c^4}\\ &=\frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\left (2 A c d^4\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}+\frac {\left (2 (b B-A c) (c d-b e)^4\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 c^4}\\ &=\frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}-\frac {2 A d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b}-\frac {2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b c^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.50, size = 212, normalized size = 0.93 \[ \frac {2 \left (\frac {(b B-A c) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt {c} \sqrt {d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )}{c^{9/2}}-105 A d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+A \sqrt {d+e x} \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )}{105 b} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2),x]

[Out]

(2*(A*Sqrt[d + e*x]*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*e^3*x^3) - 105*A*d^(7/2)*ArcTanh[Sqrt[d + e*x]/
Sqrt[d]] + ((b*B - A*c)*(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^(9/2)))/(105*b)

________________________________________________________________________________________

fricas [A]  time = 25.01, size = 1474, normalized size = 6.46 \[ \text {result too large to display} \]

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),x, algorithm="fricas")

[Out]

[1/105*(105*A*c^4*d^(7/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 105*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2
*c^2 - A*b*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A*b^3*c)*e^3)*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*(15*B*b*c^3*e^3*x^3 + 176*B*b*c^3*d^3 -
406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c - A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*
d*e^2 - 7*(B*b^2*c^2 - A*b*c^3)*e^3)*x^2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(B*b^3*c
- A*b^2*c^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), 1/105*(105*A*c^4*d^(7/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d
)/x) - 210*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A
*b^3*c)*e^3)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 2*(15*B*b*c^3*e^
3*x^3 + 176*B*b*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c - A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b
^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 - A*b*c^3)*e^3)*x^2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b
*c^3)*d*e^2 + 35*(B*b^3*c - A*b^2*c^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), 1/105*(210*A*c^4*sqrt(-d)*d^3*arctan(sq
rt(e*x + d)*sqrt(-d)/d) - 105*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)
*d*e^2 - (B*b^4 - A*b^3*c)*e^3)*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*(15*B*b*c^3*e^3*x^3 + 176*B*b*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c
- A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 - A*b*c^3)*e^3)*x^2 + (122
*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(B*b^3*c - A*b^2*c^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), 2/
105*(105*A*c^4*sqrt(-d)*d^3*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 105*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b
*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A*b^3*c)*e^3)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d
)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + (15*B*b*c^3*e^3*x^3 + 176*B*b*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*
e + 350*(B*b^3*c - A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 - A*b*c^3
)*e^3)*x^2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(B*b^3*c - A*b^2*c^2)*e^3)*x)*sqrt(e*x
+ d))/(b*c^4)]

________________________________________________________________________________________

giac [B]  time = 0.30, size = 476, normalized size = 2.09 \[ \frac {2 \, A d^{4} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d}} + \frac {2 \, {\left (B b c^{4} d^{4} - A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b c^{4}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{6} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{6} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{6} d^{2} + 105 \, \sqrt {x e + d} B c^{6} d^{3} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B b c^{5} e + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{6} e - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c^{5} d e + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{6} d e - 315 \, \sqrt {x e + d} B b c^{5} d^{2} e + 315 \, \sqrt {x e + d} A c^{6} d^{2} e + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c^{4} e^{2} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{5} e^{2} + 315 \, \sqrt {x e + d} B b^{2} c^{4} d e^{2} - 315 \, \sqrt {x e + d} A b c^{5} d e^{2} - 105 \, \sqrt {x e + d} B b^{3} c^{3} e^{3} + 105 \, \sqrt {x e + d} A b^{2} c^{4} e^{3}\right )}}{105 \, c^{7}} \]

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),x, algorithm="giac")

[Out]

2*A*d^4*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)) + 2*(B*b*c^4*d^4 - A*c^5*d^4 - 4*B*b^2*c^3*d^3*e + 4*A*b*c
^4*d^3*e + 6*B*b^3*c^2*d^2*e^2 - 6*A*b^2*c^3*d^2*e^2 - 4*B*b^4*c*d*e^3 + 4*A*b^3*c^2*d*e^3 + B*b^5*e^4 - 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*c^4) + 2/105*(15*(x*e + d)^(7/2)*
B*c^6 + 21*(x*e + d)^(5/2)*B*c^6*d + 35*(x*e + d)^(3/2)*B*c^6*d^2 + 105*sqrt(x*e + d)*B*c^6*d^3 - 21*(x*e + d)
^(5/2)*B*b*c^5*e + 21*(x*e + d)^(5/2)*A*c^6*e - 70*(x*e + d)^(3/2)*B*b*c^5*d*e + 70*(x*e + d)^(3/2)*A*c^6*d*e
- 315*sqrt(x*e + d)*B*b*c^5*d^2*e + 315*sqrt(x*e + d)*A*c^6*d^2*e + 35*(x*e + d)^(3/2)*B*b^2*c^4*e^2 - 35*(x*e
+ d)^(3/2)*A*b*c^5*e^2 + 315*sqrt(x*e + d)*B*b^2*c^4*d*e^2 - 315*sqrt(x*e + d)*A*b*c^5*d*e^2 - 105*sqrt(x*e +
d)*B*b^3*c^3*e^3 + 105*sqrt(x*e + d)*A*b^2*c^4*e^3)/c^7

________________________________________________________________________________________

maple [B]  time = 0.12, size = 741, normalized size = 3.25 \[ -\frac {2 A \,b^{3} 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 {8 A \,b^{2} 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 {12 A 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 {2 A 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}+\frac {8 A \,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}}+\frac {2 B \,b^{4} 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^{4}}-\frac {8 B \,b^{3} 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^{3}}+\frac {12 B \,b^{2} 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^{2}}-\frac {8 B 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}\, c}+\frac {2 B \,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}}-\frac {2 A \,d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b}+\frac {2 \sqrt {e x +d}\, A \,b^{2} e^{3}}{c^{3}}-\frac {6 \sqrt {e x +d}\, A b d \,e^{2}}{c^{2}}+\frac {6 \sqrt {e x +d}\, A \,d^{2} e}{c}-\frac {2 \sqrt {e x +d}\, B \,b^{3} e^{3}}{c^{4}}+\frac {6 \sqrt {e x +d}\, B \,b^{2} d \,e^{2}}{c^{3}}-\frac {6 \sqrt {e x +d}\, B b \,d^{2} e}{c^{2}}+\frac {2 \sqrt {e x +d}\, B \,d^{3}}{c}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} A b \,e^{2}}{3 c^{2}}+\frac {4 \left (e x +d \right )^{\frac {3}{2}} A d e}{3 c}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,b^{2} e^{2}}{3 c^{3}}-\frac {4 \left (e x +d \right )^{\frac {3}{2}} B b d e}{3 c^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,d^{2}}{3 c}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} A e}{5 c}-\frac {2 \left (e x +d \right )^{\frac {5}{2}} B b e}{5 c^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} B d}{5 c}+\frac {2 \left (e x +d \right )^{\frac {7}{2}} B}{7 c} \]

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),x)

[Out]

2/5/c*A*(e*x+d)^(5/2)*e+2/5/c*B*(e*x+d)^(5/2)*d+2/3/c*B*(e*x+d)^(3/2)*d^2+2/c*B*d^3*(e*x+d)^(1/2)+2/((b*e-c*d)
*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^4+8*b^2/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*e^3-12*b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2*e
^2-8*b^3/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*e^3+12*b^2/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^2*e^2-8*b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b
*e-c*d)*c)^(1/2)*c)*B*d^3*e-2/5/c^2*B*(e*x+d)^(5/2)*b*e-2/3/c^2*A*(e*x+d)^(3/2)*b*e^2+4/3/c*A*(e*x+d)^(3/2)*d*
e+2/3/c^3*B*(e*x+d)^(3/2)*b^2*e^2+2/c^3*A*b^2*e^3*(e*x+d)^(1/2)+6/c*A*d^2*e*(e*x+d)^(1/2)-2/c^4*B*b^3*e^3*(e*x
+d)^(1/2)+8/((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^4/c^4/((b*e-c*d)*c)^(1
/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*e^4-4/3/c^2*B*(e*x+d)^(3/2)*b*d*e-6/c^2*A*b*d*e^2*(e*x+d)^(1
/2)+6/c^3*B*b^2*d*e^2*(e*x+d)^(1/2)-6/c^2*B*b*d^2*e*(e*x+d)^(1/2)-2*b^3/c^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)
^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*e^4-2/b*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^
4+2/7*B*(e*x+d)^(7/2)/c-2*A*d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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),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?

________________________________________________________________________________________

mupad [B]  time = 2.36, size = 6515, normalized size = 28.57 \[ \text {result too large to display} \]

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),x)

[Out]

((2*A*e - 2*B*d)/(5*c) - (2*B*(b*e - 2*c*d))/(5*c^2))*(d + e*x)^(5/2) - (((c*d^2 - b*d*e)*((2*A*e - 2*B*d)/c -
(2*B*(b*e - 2*c*d))/c^2))/c - ((b*e - 2*c*d)*(((b*e - 2*c*d)*((2*A*e - 2*B*d)/c - (2*B*(b*e - 2*c*d))/c^2))/c
+ (2*B*(c*d^2 - b*d*e))/c^2))/c)*(d + e*x)^(1/2) - (((b*e - 2*c*d)*((2*A*e - 2*B*d)/c - (2*B*(b*e - 2*c*d))/c
^2))/(3*c) + (2*B*(c*d^2 - b*d*e))/(3*c^2))*(d + e*x)^(3/2) + (2*B*(d + e*x)^(7/2))/(7*c) - (A*atan(((A*((8*(d
+ e*x)^(1/2)*(B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 2*A^2*c^10*d^8*e^2 + 28*A^2*b^2*c^8*d^6*e^4 - 56*A^2*b^3*c^7
*d^5*e^5 + 70*A^2*b^4*c^6*d^4*e^6 - 56*A^2*b^5*c^5*d^3*e^7 + 28*A^2*b^6*c^4*d^2*e^8 + B^2*b^2*c^8*d^8*e^2 - 8*
B^2*b^3*c^7*d^7*e^3 + 28*B^2*b^4*c^6*d^6*e^4 - 56*B^2*b^5*c^5*d^5*e^5 + 70*B^2*b^6*c^4*d^4*e^6 - 56*B^2*b^7*c^
3*d^3*e^7 + 28*B^2*b^8*c^2*d^2*e^8 - 8*B^2*b^9*c*d*e^9 - 8*A^2*b*c^9*d^7*e^3 - 8*A^2*b^7*c^3*d*e^9 - 2*A*B*b^9
*c*e^10 - 2*A*B*b*c^9*d^8*e^2 + 16*A*B*b^8*c^2*d*e^9 + 16*A*B*b^2*c^8*d^7*e^3 - 56*A*B*b^3*c^7*d^6*e^4 + 112*A
*B*b^4*c^6*d^5*e^5 - 140*A*B*b^5*c^5*d^4*e^6 + 112*A*B*b^6*c^4*d^3*e^7 - 56*A*B*b^7*c^3*d^2*e^8))/c^7 + (A*(d^
7)^(1/2)*((8*(B*b^6*c^5*d*e^6 - A*b^5*c^6*d*e^6 + 3*A*b^2*c^9*d^4*e^3 - 6*A*b^3*c^8*d^3*e^4 + 4*A*b^4*c^7*d^2*
e^5 + B*b^2*c^9*d^5*e^2 - 4*B*b^3*c^8*d^4*e^3 + 6*B*b^4*c^7*d^3*e^4 - 4*B*b^5*c^6*d^2*e^5))/c^7 + (8*A*(b^3*c^
9*e^3 - 2*b^2*c^10*d*e^2)*(d^7)^(1/2)*(d + e*x)^(1/2))/(b*c^7)))/b)*(d^7)^(1/2)*1i)/b + (A*((8*(d + e*x)^(1/2)
*(B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 2*A^2*c^10*d^8*e^2 + 28*A^2*b^2*c^8*d^6*e^4 - 56*A^2*b^3*c^7*d^5*e^5 + 70
*A^2*b^4*c^6*d^4*e^6 - 56*A^2*b^5*c^5*d^3*e^7 + 28*A^2*b^6*c^4*d^2*e^8 + B^2*b^2*c^8*d^8*e^2 - 8*B^2*b^3*c^7*d
^7*e^3 + 28*B^2*b^4*c^6*d^6*e^4 - 56*B^2*b^5*c^5*d^5*e^5 + 70*B^2*b^6*c^4*d^4*e^6 - 56*B^2*b^7*c^3*d^3*e^7 + 2
8*B^2*b^8*c^2*d^2*e^8 - 8*B^2*b^9*c*d*e^9 - 8*A^2*b*c^9*d^7*e^3 - 8*A^2*b^7*c^3*d*e^9 - 2*A*B*b^9*c*e^10 - 2*A
*B*b*c^9*d^8*e^2 + 16*A*B*b^8*c^2*d*e^9 + 16*A*B*b^2*c^8*d^7*e^3 - 56*A*B*b^3*c^7*d^6*e^4 + 112*A*B*b^4*c^6*d^
5*e^5 - 140*A*B*b^5*c^5*d^4*e^6 + 112*A*B*b^6*c^4*d^3*e^7 - 56*A*B*b^7*c^3*d^2*e^8))/c^7 - (A*(d^7)^(1/2)*((8*
(B*b^6*c^5*d*e^6 - A*b^5*c^6*d*e^6 + 3*A*b^2*c^9*d^4*e^3 - 6*A*b^3*c^8*d^3*e^4 + 4*A*b^4*c^7*d^2*e^5 + B*b^2*c
^9*d^5*e^2 - 4*B*b^3*c^8*d^4*e^3 + 6*B*b^4*c^7*d^3*e^4 - 4*B*b^5*c^6*d^2*e^5))/c^7 - (8*A*(b^3*c^9*e^3 - 2*b^2
*c^10*d*e^2)*(d^7)^(1/2)*(d + e*x)^(1/2))/(b*c^7)))/b)*(d^7)^(1/2)*1i)/b)/((16*(4*A^3*c^9*d^11*e^3 + 52*A^3*b^
2*c^7*d^9*e^5 - 69*A^3*b^3*c^6*d^8*e^6 + 56*A^3*b^4*c^5*d^7*e^7 - 28*A^3*b^5*c^4*d^6*e^8 + 8*A^3*b^6*c^3*d^5*e
^9 - A^3*b^7*c^2*d^4*e^10 - A*B^2*b^9*d^4*e^10 + A^2*B*c^9*d^12*e^2 - 22*A^3*b*c^8*d^10*e^4 + 8*A*B^2*b^2*c^7*
d^11*e^3 - 28*A*B^2*b^3*c^6*d^10*e^4 + 56*A*B^2*b^4*c^5*d^9*e^5 - 70*A*B^2*b^5*c^4*d^8*e^6 + 56*A*B^2*b^6*c^3*
d^7*e^7 - 28*A*B^2*b^7*c^2*d^6*e^8 + 50*A^2*B*b^2*c^7*d^10*e^4 - 108*A^2*B*b^3*c^6*d^9*e^5 + 139*A^2*B*b^4*c^5
*d^8*e^6 - 112*A^2*B*b^5*c^4*d^7*e^7 + 56*A^2*B*b^6*c^3*d^6*e^8 - 16*A^2*B*b^7*c^2*d^5*e^9 - A*B^2*b*c^8*d^12*
e^2 + 8*A*B^2*b^8*c*d^5*e^9 - 12*A^2*B*b*c^8*d^11*e^3 + 2*A^2*B*b^8*c*d^4*e^10))/c^7 + (A*((8*(d + e*x)^(1/2)*
(B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 2*A^2*c^10*d^8*e^2 + 28*A^2*b^2*c^8*d^6*e^4 - 56*A^2*b^3*c^7*d^5*e^5 + 70*
A^2*b^4*c^6*d^4*e^6 - 56*A^2*b^5*c^5*d^3*e^7 + 28*A^2*b^6*c^4*d^2*e^8 + B^2*b^2*c^8*d^8*e^2 - 8*B^2*b^3*c^7*d^
7*e^3 + 28*B^2*b^4*c^6*d^6*e^4 - 56*B^2*b^5*c^5*d^5*e^5 + 70*B^2*b^6*c^4*d^4*e^6 - 56*B^2*b^7*c^3*d^3*e^7 + 28
*B^2*b^8*c^2*d^2*e^8 - 8*B^2*b^9*c*d*e^9 - 8*A^2*b*c^9*d^7*e^3 - 8*A^2*b^7*c^3*d*e^9 - 2*A*B*b^9*c*e^10 - 2*A*
B*b*c^9*d^8*e^2 + 16*A*B*b^8*c^2*d*e^9 + 16*A*B*b^2*c^8*d^7*e^3 - 56*A*B*b^3*c^7*d^6*e^4 + 112*A*B*b^4*c^6*d^5
*e^5 - 140*A*B*b^5*c^5*d^4*e^6 + 112*A*B*b^6*c^4*d^3*e^7 - 56*A*B*b^7*c^3*d^2*e^8))/c^7 + (A*(d^7)^(1/2)*((8*(
B*b^6*c^5*d*e^6 - A*b^5*c^6*d*e^6 + 3*A*b^2*c^9*d^4*e^3 - 6*A*b^3*c^8*d^3*e^4 + 4*A*b^4*c^7*d^2*e^5 + B*b^2*c^
9*d^5*e^2 - 4*B*b^3*c^8*d^4*e^3 + 6*B*b^4*c^7*d^3*e^4 - 4*B*b^5*c^6*d^2*e^5))/c^7 + (8*A*(b^3*c^9*e^3 - 2*b^2*
c^10*d*e^2)*(d^7)^(1/2)*(d + e*x)^(1/2))/(b*c^7)))/b)*(d^7)^(1/2))/b - (A*((8*(d + e*x)^(1/2)*(B^2*b^10*e^10 +
A^2*b^8*c^2*e^10 + 2*A^2*c^10*d^8*e^2 + 28*A^2*b^2*c^8*d^6*e^4 - 56*A^2*b^3*c^7*d^5*e^5 + 70*A^2*b^4*c^6*d^4*
e^6 - 56*A^2*b^5*c^5*d^3*e^7 + 28*A^2*b^6*c^4*d^2*e^8 + B^2*b^2*c^8*d^8*e^2 - 8*B^2*b^3*c^7*d^7*e^3 + 28*B^2*b
^4*c^6*d^6*e^4 - 56*B^2*b^5*c^5*d^5*e^5 + 70*B^2*b^6*c^4*d^4*e^6 - 56*B^2*b^7*c^3*d^3*e^7 + 28*B^2*b^8*c^2*d^2
*e^8 - 8*B^2*b^9*c*d*e^9 - 8*A^2*b*c^9*d^7*e^3 - 8*A^2*b^7*c^3*d*e^9 - 2*A*B*b^9*c*e^10 - 2*A*B*b*c^9*d^8*e^2
+ 16*A*B*b^8*c^2*d*e^9 + 16*A*B*b^2*c^8*d^7*e^3 - 56*A*B*b^3*c^7*d^6*e^4 + 112*A*B*b^4*c^6*d^5*e^5 - 140*A*B*b
^5*c^5*d^4*e^6 + 112*A*B*b^6*c^4*d^3*e^7 - 56*A*B*b^7*c^3*d^2*e^8))/c^7 - (A*(d^7)^(1/2)*((8*(B*b^6*c^5*d*e^6
- A*b^5*c^6*d*e^6 + 3*A*b^2*c^9*d^4*e^3 - 6*A*b^3*c^8*d^3*e^4 + 4*A*b^4*c^7*d^2*e^5 + B*b^2*c^9*d^5*e^2 - 4*B*
b^3*c^8*d^4*e^3 + 6*B*b^4*c^7*d^3*e^4 - 4*B*b^5*c^6*d^2*e^5))/c^7 - (8*A*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2)*(d^7
)^(1/2)*(d + e*x)^(1/2))/(b*c^7)))/b)*(d^7)^(1/2))/b))*(d^7)^(1/2)*2i)/b - (atan((((-c^9*(b*e - c*d)^7)^(1/2)*
((8*(d + e*x)^(1/2)*(B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 2*A^2*c^10*d^8*e^2 + 28*A^2*b^2*c^8*d^6*e^4 - 56*A^2*b
^3*c^7*d^5*e^5 + 70*A^2*b^4*c^6*d^4*e^6 - 56*A^2*b^5*c^5*d^3*e^7 + 28*A^2*b^6*c^4*d^2*e^8 + B^2*b^2*c^8*d^8*e^
2 - 8*B^2*b^3*c^7*d^7*e^3 + 28*B^2*b^4*c^6*d^6*e^4 - 56*B^2*b^5*c^5*d^5*e^5 + 70*B^2*b^6*c^4*d^4*e^6 - 56*B^2*
b^7*c^3*d^3*e^7 + 28*B^2*b^8*c^2*d^2*e^8 - 8*B^2*b^9*c*d*e^9 - 8*A^2*b*c^9*d^7*e^3 - 8*A^2*b^7*c^3*d*e^9 - 2*A
*B*b^9*c*e^10 - 2*A*B*b*c^9*d^8*e^2 + 16*A*B*b^8*c^2*d*e^9 + 16*A*B*b^2*c^8*d^7*e^3 - 56*A*B*b^3*c^7*d^6*e^4 +
112*A*B*b^4*c^6*d^5*e^5 - 140*A*B*b^5*c^5*d^4*e^6 + 112*A*B*b^6*c^4*d^3*e^7 - 56*A*B*b^7*c^3*d^2*e^8))/c^7 +
((-c^9*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*((8*(B*b^6*c^5*d*e^6 - A*b^5*c^6*d*e^6 + 3*A*b^2*c^9*d^4*e^3 - 6*A*b^3
*c^8*d^3*e^4 + 4*A*b^4*c^7*d^2*e^5 + B*b^2*c^9*d^5*e^2 - 4*B*b^3*c^8*d^4*e^3 + 6*B*b^4*c^7*d^3*e^4 - 4*B*b^5*c
^6*d^2*e^5))/c^7 + (8*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2)*(-c^9*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2))
/(b*c^16)))/(b*c^9))*(A*c - B*b)*1i)/(b*c^9) + ((-c^9*(b*e - c*d)^7)^(1/2)*((8*(d + e*x)^(1/2)*(B^2*b^10*e^10
+ A^2*b^8*c^2*e^10 + 2*A^2*c^10*d^8*e^2 + 28*A^2*b^2*c^8*d^6*e^4 - 56*A^2*b^3*c^7*d^5*e^5 + 70*A^2*b^4*c^6*d^4
*e^6 - 56*A^2*b^5*c^5*d^3*e^7 + 28*A^2*b^6*c^4*d^2*e^8 + B^2*b^2*c^8*d^8*e^2 - 8*B^2*b^3*c^7*d^7*e^3 + 28*B^2*
b^4*c^6*d^6*e^4 - 56*B^2*b^5*c^5*d^5*e^5 + 70*B^2*b^6*c^4*d^4*e^6 - 56*B^2*b^7*c^3*d^3*e^7 + 28*B^2*b^8*c^2*d^
2*e^8 - 8*B^2*b^9*c*d*e^9 - 8*A^2*b*c^9*d^7*e^3 - 8*A^2*b^7*c^3*d*e^9 - 2*A*B*b^9*c*e^10 - 2*A*B*b*c^9*d^8*e^2
+ 16*A*B*b^8*c^2*d*e^9 + 16*A*B*b^2*c^8*d^7*e^3 - 56*A*B*b^3*c^7*d^6*e^4 + 112*A*B*b^4*c^6*d^5*e^5 - 140*A*B*
b^5*c^5*d^4*e^6 + 112*A*B*b^6*c^4*d^3*e^7 - 56*A*B*b^7*c^3*d^2*e^8))/c^7 - ((-c^9*(b*e - c*d)^7)^(1/2)*(A*c -
B*b)*((8*(B*b^6*c^5*d*e^6 - A*b^5*c^6*d*e^6 + 3*A*b^2*c^9*d^4*e^3 - 6*A*b^3*c^8*d^3*e^4 + 4*A*b^4*c^7*d^2*e^5
+ B*b^2*c^9*d^5*e^2 - 4*B*b^3*c^8*d^4*e^3 + 6*B*b^4*c^7*d^3*e^4 - 4*B*b^5*c^6*d^2*e^5))/c^7 - (8*(b^3*c^9*e^3
- 2*b^2*c^10*d*e^2)*(-c^9*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2))/(b*c^16)))/(b*c^9))*(A*c - B*b)*1i
)/(b*c^9))/((16*(4*A^3*c^9*d^11*e^3 + 52*A^3*b^2*c^7*d^9*e^5 - 69*A^3*b^3*c^6*d^8*e^6 + 56*A^3*b^4*c^5*d^7*e^7
- 28*A^3*b^5*c^4*d^6*e^8 + 8*A^3*b^6*c^3*d^5*e^9 - A^3*b^7*c^2*d^4*e^10 - A*B^2*b^9*d^4*e^10 + A^2*B*c^9*d^12
*e^2 - 22*A^3*b*c^8*d^10*e^4 + 8*A*B^2*b^2*c^7*d^11*e^3 - 28*A*B^2*b^3*c^6*d^10*e^4 + 56*A*B^2*b^4*c^5*d^9*e^5
- 70*A*B^2*b^5*c^4*d^8*e^6 + 56*A*B^2*b^6*c^3*d^7*e^7 - 28*A*B^2*b^7*c^2*d^6*e^8 + 50*A^2*B*b^2*c^7*d^10*e^4
- 108*A^2*B*b^3*c^6*d^9*e^5 + 139*A^2*B*b^4*c^5*d^8*e^6 - 112*A^2*B*b^5*c^4*d^7*e^7 + 56*A^2*B*b^6*c^3*d^6*e^8
- 16*A^2*B*b^7*c^2*d^5*e^9 - A*B^2*b*c^8*d^12*e^2 + 8*A*B^2*b^8*c*d^5*e^9 - 12*A^2*B*b*c^8*d^11*e^3 + 2*A^2*B
*b^8*c*d^4*e^10))/c^7 + ((-c^9*(b*e - c*d)^7)^(1/2)*((8*(d + e*x)^(1/2)*(B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 2*
A^2*c^10*d^8*e^2 + 28*A^2*b^2*c^8*d^6*e^4 - 56*A^2*b^3*c^7*d^5*e^5 + 70*A^2*b^4*c^6*d^4*e^6 - 56*A^2*b^5*c^5*d
^3*e^7 + 28*A^2*b^6*c^4*d^2*e^8 + B^2*b^2*c^8*d^8*e^2 - 8*B^2*b^3*c^7*d^7*e^3 + 28*B^2*b^4*c^6*d^6*e^4 - 56*B^
2*b^5*c^5*d^5*e^5 + 70*B^2*b^6*c^4*d^4*e^6 - 56*B^2*b^7*c^3*d^3*e^7 + 28*B^2*b^8*c^2*d^2*e^8 - 8*B^2*b^9*c*d*e
^9 - 8*A^2*b*c^9*d^7*e^3 - 8*A^2*b^7*c^3*d*e^9 - 2*A*B*b^9*c*e^10 - 2*A*B*b*c^9*d^8*e^2 + 16*A*B*b^8*c^2*d*e^9
+ 16*A*B*b^2*c^8*d^7*e^3 - 56*A*B*b^3*c^7*d^6*e^4 + 112*A*B*b^4*c^6*d^5*e^5 - 140*A*B*b^5*c^5*d^4*e^6 + 112*A
*B*b^6*c^4*d^3*e^7 - 56*A*B*b^7*c^3*d^2*e^8))/c^7 + ((-c^9*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*((8*(B*b^6*c^5*d*e
^6 - A*b^5*c^6*d*e^6 + 3*A*b^2*c^9*d^4*e^3 - 6*A*b^3*c^8*d^3*e^4 + 4*A*b^4*c^7*d^2*e^5 + B*b^2*c^9*d^5*e^2 - 4
*B*b^3*c^8*d^4*e^3 + 6*B*b^4*c^7*d^3*e^4 - 4*B*b^5*c^6*d^2*e^5))/c^7 + (8*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2)*(-c
^9*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2))/(b*c^16)))/(b*c^9))*(A*c - B*b))/(b*c^9) - ((-c^9*(b*e -
c*d)^7)^(1/2)*((8*(d + e*x)^(1/2)*(B^2*b^10*e^10 + A^2*b^8*c^2*e^10 + 2*A^2*c^10*d^8*e^2 + 28*A^2*b^2*c^8*d^6*
e^4 - 56*A^2*b^3*c^7*d^5*e^5 + 70*A^2*b^4*c^6*d^4*e^6 - 56*A^2*b^5*c^5*d^3*e^7 + 28*A^2*b^6*c^4*d^2*e^8 + B^2*
b^2*c^8*d^8*e^2 - 8*B^2*b^3*c^7*d^7*e^3 + 28*B^2*b^4*c^6*d^6*e^4 - 56*B^2*b^5*c^5*d^5*e^5 + 70*B^2*b^6*c^4*d^4
*e^6 - 56*B^2*b^7*c^3*d^3*e^7 + 28*B^2*b^8*c^2*d^2*e^8 - 8*B^2*b^9*c*d*e^9 - 8*A^2*b*c^9*d^7*e^3 - 8*A^2*b^7*c
^3*d*e^9 - 2*A*B*b^9*c*e^10 - 2*A*B*b*c^9*d^8*e^2 + 16*A*B*b^8*c^2*d*e^9 + 16*A*B*b^2*c^8*d^7*e^3 - 56*A*B*b^3
*c^7*d^6*e^4 + 112*A*B*b^4*c^6*d^5*e^5 - 140*A*B*b^5*c^5*d^4*e^6 + 112*A*B*b^6*c^4*d^3*e^7 - 56*A*B*b^7*c^3*d^
2*e^8))/c^7 - ((-c^9*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*((8*(B*b^6*c^5*d*e^6 - A*b^5*c^6*d*e^6 + 3*A*b^2*c^9*d^4
*e^3 - 6*A*b^3*c^8*d^3*e^4 + 4*A*b^4*c^7*d^2*e^5 + B*b^2*c^9*d^5*e^2 - 4*B*b^3*c^8*d^4*e^3 + 6*B*b^4*c^7*d^3*e
^4 - 4*B*b^5*c^6*d^2*e^5))/c^7 - (8*(b^3*c^9*e^3 - 2*b^2*c^10*d*e^2)*(-c^9*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(d
+ e*x)^(1/2))/(b*c^16)))/(b*c^9))*(A*c - B*b))/(b*c^9)))*(-c^9*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*2i)/(b*c^9)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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),x)

[Out]

Timed out

________________________________________________________________________________________