\(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{d+e x} \, dx\) [104]

Optimal result

Integrand size = 26, antiderivative size = 410 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )\right ) \sqrt {b x+c x^2}}{64 c^2 e^4}-\frac {\left (8 A c e (6 c d-7 b e)-B \left (48 c^2 d^2-56 b c d e+3 b^2 e^2\right )\right ) x \sqrt {b x+c x^2}}{96 c e^3}-\frac {(8 B c d-3 b B e-8 A c e) x^2 \sqrt {b x+c x^2}}{24 e^2}+\frac {B x \left (b x+c x^2\right )^{3/2}}{4 e}-\frac {\left (8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )-B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{5/2} e^5}-\frac {2 d^{3/2} (B d-A e) (c d-b e)^{3/2} \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{e^5} \] Output:

1/64*(8*A*c*e*(b^2*e^2-10*b*c*d*e+8*c^2*d^2)-B*(3*b^3*e^3+8*b^2*c*d*e^2-80 
*b*c^2*d^2*e+64*c^3*d^3))*(c*x^2+b*x)^(1/2)/c^2/e^4-1/96*(8*A*c*e*(-7*b*e+ 
6*c*d)-B*(3*b^2*e^2-56*b*c*d*e+48*c^2*d^2))*x*(c*x^2+b*x)^(1/2)/c/e^3-1/24 
*(-8*A*c*e-3*B*b*e+8*B*c*d)*x^2*(c*x^2+b*x)^(1/2)/e^2+1/4*B*x*(c*x^2+b*x)^ 
(3/2)/e-1/64*(8*A*c*e*(b^3*e^3+6*b^2*c*d*e^2-24*b*c^2*d^2*e+16*c^3*d^3)-B* 
(3*b^4*e^4+8*b^3*c*d*e^3+48*b^2*c^2*d^2*e^2-192*b*c^3*d^3*e+128*c^4*d^4))* 
arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/c^(5/2)/e^5-2*d^(3/2)*(-A*e+B*d)*(-b* 
e+c*d)^(3/2)*arctanh((-b*e+c*d)^(1/2)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/e^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.93 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {(x (b+c x))^{3/2} \left (\sqrt {c} e \sqrt {x} \sqrt {b+c x} \left (8 A c e \left (3 b^2 e^2+2 b c e (-15 d+7 e x)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+B \left (-9 b^3 e^3+6 b^2 c e^2 (-4 d+e x)+8 b c^2 e \left (30 d^2-14 d e x+9 e^2 x^2\right )-16 c^3 \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )\right )\right )+384 c^{3/2} \sqrt {d} (B d-A e) (c d-b e) \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+384 c^{3/2} \sqrt {d} (B d-A e) (c d-b e) \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+6 \left (-8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )+B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{192 c^{5/2} e^5 x^{3/2} (b+c x)^{3/2}} \] Input:

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x),x]
 

Output:

((x*(b + c*x))^(3/2)*(Sqrt[c]*e*Sqrt[x]*Sqrt[b + c*x]*(8*A*c*e*(3*b^2*e^2 
+ 2*b*c*e*(-15*d + 7*e*x) + 4*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + B*(-9*b 
^3*e^3 + 6*b^2*c*e^2*(-4*d + e*x) + 8*b*c^2*e*(30*d^2 - 14*d*e*x + 9*e^2*x 
^2) - 16*c^3*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3*x^3))) + 384*c^(3/2 
)*Sqrt[d]*(B*d - A*e)*(c*d - b*e)*(c*d - b*e - I*Sqrt[b]*Sqrt[e]*Sqrt[c*d 
- b*e])*Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTa 
n[(Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/( 
Sqrt[d]*(-Sqrt[b] + Sqrt[b + c*x]))] + 384*c^(3/2)*Sqrt[d]*(B*d - A*e)*(c* 
d - b*e)*(c*d - b*e + I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*b 
*e + (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTan[(Sqrt[-(c*d) + 2*b*e + 
(2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqrt[d]*(-Sqrt[b] + Sqrt[ 
b + c*x]))] + 6*(-8*A*c*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b 
^3*e^3) + B*(128*c^4*d^4 - 192*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 + 8*b^3*c* 
d*e^3 + 3*b^4*e^4))*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]) 
)/(192*c^(5/2)*e^5*x^(3/2)*(b + c*x)^(3/2))
 

Rubi [A] (verified)

Time = 1.57 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1231, 27, 1231, 27, 1269, 1091, 219, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx\)

\(\Big \downarrow \) 1231

\(\displaystyle -\frac {\int -\frac {\left (b d (8 B c d-3 b B e-8 A c e)-\left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c e d-3 b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)}dx}{8 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (b d (8 B c d-3 b B e-8 A c e)-\left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c e d-3 b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{d+e x}dx}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\int \frac {b d \left (8 A c e \left (8 c^2 d^2-10 b c e d+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 e d^2+8 b^2 c e^2 d+3 b^3 e^3\right )\right )+\left (4 b c d e (2 c d-b e) (8 B c d-3 b B e-8 A c e)+\left (8 c^2 d^2-4 b c e d-b^2 e^2\right ) \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c e d-3 b^2 e^2\right )\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x}}dx}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\int \frac {b d \left (8 A c e \left (8 c^2 d^2-10 b c e d+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 e d^2+8 b^2 c e^2 d+3 b^3 e^3\right )\right )+\left (4 b c d e (2 c d-b e) (8 B c d-3 b B e-8 A c e)+\left (8 c^2 d^2-4 b c e d-b^2 e^2\right ) \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c e d-3 b^2 e^2\right )\right )\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {\left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}+\frac {128 c^2 d^2 (B d-A e) (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}+\frac {128 c^2 d^2 (B d-A e) (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {128 c^2 d^2 (B d-A e) (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right )}{\sqrt {c} e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right )}{\sqrt {c} e}-\frac {256 c^2 d^2 (B d-A e) (c d-b e)^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right )}{\sqrt {c} e}+\frac {128 c^2 d^{3/2} (B d-A e) (c d-b e)^{3/2} \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\)

Input:

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x),x]
 

Output:

-1/24*((8*B*c*d - 3*b*B*e - 8*A*c*e - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(c*e 
^2) + (((8*A*c*e*(8*c^2*d^2 - 10*b*c*d*e + b^2*e^2) - B*(64*c^3*d^3 - 80*b 
*c^2*d^2*e + 8*b^2*c*d*e^2 + 3*b^3*e^3) - 2*c*e*(8*A*c*e*(2*c*d - b*e) - B 
*(16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(4*c*e^2) - ( 
(2*(4*b*c*d*e*(2*c*d - b*e)*(8*B*c*d - 3*b*B*e - 8*A*c*e) + (8*c^2*d^2 - 4 
*b*c*d*e - b^2*e^2)*(8*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 8*b*c*d*e - 3 
*b^2*e^2)))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(Sqrt[c]*e) + (128*c^2 
*d^(3/2)*(B*d - A*e)*(c*d - b*e)^(3/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2* 
Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e)/(8*c*e^2))/(16*c*e^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {-16 d^{2} \left (c^{\frac {5}{2}} b^{2} e^{2}-2 c^{\frac {7}{2}} b d e +c^{\frac {9}{2}} d^{2}\right ) \left (A e -B d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\left (\left (\left (-A \,b^{3} c +\frac {3}{8} B \,b^{4}\right ) e^{4}+\left (-6 A \,b^{2} c^{2}+B \,b^{3} c \right ) d \,e^{3}+6 \left (4 A b \,c^{3}+B \,b^{2} c^{2}\right ) d^{2} e^{2}-16 c^{3} d^{3} \left (A c +\frac {3 B b}{2}\right ) e +16 B \,c^{4} d^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+e \sqrt {x \left (c x +b \right )}\, \left (2 \left (\left (\frac {4}{3} A \,x^{2}+B \,x^{3}\right ) e^{3}-2 d x \left (\frac {2 B x}{3}+A \right ) e^{2}+4 e \left (\frac {B x}{2}+A \right ) d^{2}-4 B \,d^{3}\right ) c^{\frac {7}{2}}+e \left (\left (\left (3 B \,x^{2}+\frac {14}{3} A x \right ) e^{2}-10 d \left (\frac {7 B x}{15}+A \right ) e +10 B \,d^{2}\right ) c^{\frac {5}{2}}+\left (\left (\left (\frac {B x}{4}+A \right ) e -B d \right ) c^{\frac {3}{2}}-\frac {3 B b e \sqrt {c}}{8}\right ) e b \right ) b \right )\right ) \sqrt {d \left (b e -c d \right )}}{8 c^{\frac {5}{2}} \sqrt {d \left (b e -c d \right )}\, e^{5}}\)	\(344\)
risch	\(\frac {\left (48 B \,c^{3} e^{3} x^{3}+64 A \,c^{3} e^{3} x^{2}+72 B b \,c^{2} e^{3} x^{2}-64 B \,c^{3} d \,e^{2} x^{2}+112 A b \,c^{2} e^{3} x -96 A \,c^{3} d \,e^{2} x +6 B \,b^{2} c \,e^{3} x -112 B b \,c^{2} d \,e^{2} x +96 B \,c^{3} d^{2} e x +24 A \,b^{2} c \,e^{3}-240 A b \,c^{2} d \,e^{2}+192 A \,c^{3} d^{2} e -9 B \,e^{3} b^{3}-24 B \,b^{2} c d \,e^{2}+240 B b \,c^{2} d^{2} e -192 B \,c^{3} d^{3}\right ) x \left (c x +b \right )}{192 c^{2} e^{4} \sqrt {x \left (c x +b \right )}}-\frac {\frac {\left (8 A \,b^{3} c \,e^{4}+48 A \,b^{2} c^{2} d \,e^{3}-192 A b \,c^{3} d^{2} e^{2}+128 A \,c^{4} d^{3} e -3 B \,b^{4} e^{4}-8 B \,b^{3} c d \,e^{3}-48 B \,b^{2} c^{2} d^{2} e^{2}+192 B b \,c^{3} d^{3} e -128 B \,c^{4} d^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}+\frac {128 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right ) c^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{128 e^{4} c^{2}}\)	\(537\)
default	\(\frac {B \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{e}+\frac {\left (A e -B d \right ) \left (\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}-\frac {d \left (b e -c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}\right )}{e^{2}}\)	\(649\)

Input:

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/8/c^(5/2)/(d*(b*e-c*d))^(1/2)*(-16*d^2*(c^(5/2)*b^2*e^2-2*c^(7/2)*b*d*e+ 
c^(9/2)*d^2)*(A*e-B*d)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+( 
((-A*b^3*c+3/8*B*b^4)*e^4+(-6*A*b^2*c^2+B*b^3*c)*d*e^3+6*(4*A*b*c^3+B*b^2* 
c^2)*d^2*e^2-16*c^3*d^3*(A*c+3/2*B*b)*e+16*B*c^4*d^4)*arctanh((x*(c*x+b))^ 
(1/2)/x/c^(1/2))+e*(x*(c*x+b))^(1/2)*(2*((4/3*A*x^2+B*x^3)*e^3-2*d*x*(2/3* 
B*x+A)*e^2+4*e*(1/2*B*x+A)*d^2-4*B*d^3)*c^(7/2)+e*(((3*B*x^2+14/3*A*x)*e^2 
-10*d*(7/15*B*x+A)*e+10*B*d^2)*c^(5/2)+(((1/4*B*x+A)*e-B*d)*c^(3/2)-3/8*B* 
b*e*c^(1/2))*e*b)*b))*(d*(b*e-c*d))^(1/2))/e^5
 

Fricas [A] (verification not implemented)

Time = 17.08 (sec) , antiderivative size = 1688, normalized size of antiderivative = 4.12 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d),x, algorithm="fricas")
 

Output:

[-1/384*(3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 
 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b 
^3*c)*e^4)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 384*(B*c 
^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*sqrt(c*d^2 - b*d*e)*log( 
(b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d 
)) - 2*(48*B*c^4*e^4*x^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)*d^2* 
e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^2)*e^4 
- 8*(8*B*c^4*d*e^3 - (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 
- 8*(7*B*b*c^3 + 6*A*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)*sqrt( 
c*x^2 + b*x))/(c^3*e^5), 1/384*(768*(B*c^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 
+ A*c^4)*d^2*e)*sqrt(-c*d^2 + b*d*e)*arctan(sqrt(-c*d^2 + b*d*e)*sqrt(c*x^ 
2 + b*x)/(c*d*x + b*d)) - 3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3* 
e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + 
 (3*B*b^4 - 8*A*b^3*c)*e^4)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sq 
rt(c)) + 2*(48*B*c^4*e^4*x^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)* 
d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^2)* 
e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2* 
e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)*s 
qrt(c*x^2 + b*x))/(c^3*e^5), -1/192*(3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2* 
A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{d + e x}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d),x)
 

Output:

Integral((x*(b + c*x))**(3/2)*(A + B*x)/(d + e*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{d+e\,x} \,d x \] Input:

int(((b*x + c*x^2)^(3/2)*(A + B*x))/(d + e*x),x)
 

Output:

int(((b*x + c*x^2)^(3/2)*(A + B*x))/(d + e*x), x)
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 1075, normalized size of antiderivative = 2.62 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d),x)
 

Output:

( - 384*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c 
*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*a*b*c**3*d*e**2 + 384*sq 
rt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt 
(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*a*c**4*d**2*e + 384*sqrt(d)*sqrt(b 
*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)* 
sqrt(c))/(sqrt(d)*sqrt(c)))*b**2*c**3*d**2*e - 384*sqrt(d)*sqrt(b*e - c*d) 
*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/ 
(sqrt(d)*sqrt(c)))*b*c**4*d**3 - 384*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b* 
e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt( 
c)))*a*b*c**3*d*e**2 + 384*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + 
 sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*a*c** 
4*d**2*e + 384*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqr 
t(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**2*c**3*d**2*e 
- 384*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x 
) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*c**4*d**3 + 24*sqrt(x)*s 
qrt(b + c*x)*a*b**2*c**2*e**4 - 240*sqrt(x)*sqrt(b + c*x)*a*b*c**3*d*e**3 
+ 112*sqrt(x)*sqrt(b + c*x)*a*b*c**3*e**4*x + 192*sqrt(x)*sqrt(b + c*x)*a* 
c**4*d**2*e**2 - 96*sqrt(x)*sqrt(b + c*x)*a*c**4*d*e**3*x + 64*sqrt(x)*sqr 
t(b + c*x)*a*c**4*e**4*x**2 - 9*sqrt(x)*sqrt(b + c*x)*b**4*c*e**4 - 24*sqr 
t(x)*sqrt(b + c*x)*b**3*c**2*d*e**3 + 6*sqrt(x)*sqrt(b + c*x)*b**3*c**2...