\(\int (A+B x) (d+e x) (b x+c x^2)^{3/2} \, dx\) [103]

Optimal result

Integrand size = 24, antiderivative size = 317 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=-\frac {b^3 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) \sqrt {b x+c x^2}}{512 c^4}+\frac {b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) x \sqrt {b x+c x^2}}{768 c^3}+\frac {b \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) x^2 \sqrt {b x+c x^2}}{64 c^2}+\frac {\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) x^3 \sqrt {b x+c x^2}}{96 c}+\frac {(12 B c d-7 b B e+12 A c e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {B e x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}} \] Output:

-1/512*b^3*(24*A*c^2*d+7*b^2*B*e-12*b*c*(A*e+B*d))*(c*x^2+b*x)^(1/2)/c^4+1 
/768*b^2*(24*A*c^2*d+7*b^2*B*e-12*b*c*(A*e+B*d))*x*(c*x^2+b*x)^(1/2)/c^3+1 
/64*b*(24*A*c^2*d+7*b^2*B*e-12*b*c*(A*e+B*d))*x^2*(c*x^2+b*x)^(1/2)/c^2+1/ 
96*(24*A*c^2*d+7*b^2*B*e-12*b*c*(A*e+B*d))*x^3*(c*x^2+b*x)^(1/2)/c+1/60*(1 
2*A*c*e-7*B*b*e+12*B*c*d)*(c*x^2+b*x)^(5/2)/c^2+1/6*B*e*x*(c*x^2+b*x)^(5/2 
)/c+1/512*b^4*(24*A*c^2*d+7*b^2*B*e-12*b*c*(A*e+B*d))*arctanh(c^(1/2)*x/(c 
*x^2+b*x)^(1/2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 2.49 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.95 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (-105 b^5 B e+10 b^4 c (18 B d+18 A e+7 B e x)+48 b^2 c^3 x (B x (2 d+e x)+A (5 d+2 e x))+128 c^5 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))-8 b^3 c^2 (15 A (3 d+e x)+B x (15 d+7 e x))+64 b c^4 x^2 (B x (33 d+26 e x)+A (45 d+33 e x))\right )+360 b^5 c (B d+A e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+30 b^4 \left (24 A c^2 d+7 b^2 B e\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{7680 c^{9/2} \sqrt {x (b+c x)}} \] Input:

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

Output:

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(-105*b^5*B*e + 10*b 
^4*c*(18*B*d + 18*A*e + 7*B*e*x) + 48*b^2*c^3*x*(B*x*(2*d + e*x) + A*(5*d 
+ 2*e*x)) + 128*c^5*x^3*(3*A*(5*d + 4*e*x) + 2*B*x*(6*d + 5*e*x)) - 8*b^3* 
c^2*(15*A*(3*d + e*x) + B*x*(15*d + 7*e*x)) + 64*b*c^4*x^2*(B*x*(33*d + 26 
*e*x) + A*(45*d + 33*e*x))) + 360*b^5*c*(B*d + A*e)*ArcTanh[(Sqrt[c]*Sqrt[ 
x])/(Sqrt[b] - Sqrt[b + c*x])] + 30*b^4*(24*A*c^2*d + 7*b^2*B*e)*ArcTanh[( 
Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(7680*c^(9/2)*Sqrt[x*(b + c 
*x)])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.55, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1225, 1087, 1087, 1091, 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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right ) \int \left (c x^2+b x\right )^{3/2}dx}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^2+b x}dx}{16 c}\right )}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\right )}{16 c}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{24 c^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2}\)

Input:

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

Output:

-1/60*((7*b*B*e - 12*c*(B*d + A*e) - 10*B*c*e*x)*(b*x + c*x^2)^(5/2))/c^2 
+ ((24*A*c^2*d + 7*b^2*B*e - 12*b*c*(B*d + A*e))*(((b + 2*c*x)*(b*x + c*x^ 
2)^(3/2))/(8*c) - (3*b^2*(((b + 2*c*x)*Sqrt[b*x + c*x^2])/(4*c) - (b^2*Arc 
Tanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(3/2))))/(16*c)))/(24*c^2)
 

Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.67

method	result	size
pseudoelliptic	\(\frac {-\frac {3 \left (-\frac {7 b^{2} B e}{12}+b c \left (A e +B d \right )-2 A \,c^{2} d \right ) b^{4} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{128}+\frac {3 \left (\frac {32 \left (\frac {2 B e \,x^{2}}{3}+\frac {4 \left (A e +B d \right ) x}{5}+A d \right ) x^{3} c^{\frac {11}{2}}}{3}+b \left (-2 \left (\frac {7 B e \,x^{2}}{45}+\frac {\left (A e +B d \right ) x}{3}+A d \right ) b^{2} c^{\frac {5}{2}}+\frac {4 \left (\frac {B e \,x^{2}}{5}+\frac {2 \left (A e +B d \right ) x}{5}+A d \right ) x b \,c^{\frac {7}{2}}}{3}+16 x^{2} \left (\frac {26 B e \,x^{2}}{45}+\frac {11 \left (A e +B d \right ) x}{15}+A d \right ) c^{\frac {9}{2}}+\left (\left (\frac {7}{18} B e x +A e +B d \right ) c^{\frac {3}{2}}-\frac {7 B b e \sqrt {c}}{12}\right ) b^{3}\right )\right ) \sqrt {x \left (c x +b \right )}}{128}}{c^{\frac {9}{2}}}\)	\(212\)
risch	\(\frac {\left (1280 B \,c^{5} e \,x^{5}+1536 A \,c^{5} e \,x^{4}+1664 B b \,c^{4} e \,x^{4}+1536 B \,c^{5} d \,x^{4}+2112 A b \,c^{4} e \,x^{3}+1920 A \,c^{5} d \,x^{3}+48 B \,b^{2} c^{3} e \,x^{3}+2112 B b \,c^{4} d \,x^{3}+96 A \,b^{2} c^{3} e \,x^{2}+2880 A b \,c^{4} d \,x^{2}-56 B \,b^{3} c^{2} e \,x^{2}+96 B \,b^{2} c^{3} d \,x^{2}-120 A \,b^{3} c^{2} e x +240 A \,b^{2} c^{3} d x +70 B \,b^{4} c e x -120 B \,b^{3} c^{2} d x +180 A \,b^{4} c e -360 A \,b^{3} c^{2} d -105 B \,b^{5} e +180 B \,b^{4} c d \right ) x \left (c x +b \right )}{7680 c^{4} \sqrt {x \left (c x +b \right )}}-\frac {b^{4} \left (12 A c e b -24 A \,c^{2} d -7 b^{2} B e +12 B b c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}\)	\(293\)
default	\(A d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+\left (A e +B d \right ) \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )+B e \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )\)	\(344\)

Input:

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

Output:

3/128/c^(9/2)*(-(-7/12*b^2*B*e+b*c*(A*e+B*d)-2*A*c^2*d)*b^4*arctanh((x*(c* 
x+b))^(1/2)/x/c^(1/2))+(32/3*(2/3*B*e*x^2+4/5*(A*e+B*d)*x+A*d)*x^3*c^(11/2 
)+b*(-2*(7/45*B*e*x^2+1/3*(A*e+B*d)*x+A*d)*b^2*c^(5/2)+4/3*(1/5*B*e*x^2+2/ 
5*(A*e+B*d)*x+A*d)*x*b*c^(7/2)+16*x^2*(26/45*B*e*x^2+11/15*(A*e+B*d)*x+A*d 
)*c^(9/2)+((7/18*B*e*x+A*e+B*d)*c^(3/2)-7/12*B*b*e*c^(1/2))*b^3))*(x*(c*x+ 
b))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.91 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\left [\frac {15 \, {\left (12 \, {\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} d - {\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (1280 \, B c^{6} e x^{5} + 128 \, {\left (12 \, B c^{6} d + {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} e\right )} x^{4} + 48 \, {\left (4 \, {\left (11 \, B b c^{5} + 10 \, A c^{6}\right )} d + {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} e\right )} x^{3} + 8 \, {\left (12 \, {\left (B b^{2} c^{4} + 30 \, A b c^{5}\right )} d - {\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} e\right )} x^{2} + 180 \, {\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d - 15 \, {\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )} e - 10 \, {\left (12 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d - {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{15360 \, c^{5}}, \frac {15 \, {\left (12 \, {\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} d - {\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) + {\left (1280 \, B c^{6} e x^{5} + 128 \, {\left (12 \, B c^{6} d + {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} e\right )} x^{4} + 48 \, {\left (4 \, {\left (11 \, B b c^{5} + 10 \, A c^{6}\right )} d + {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} e\right )} x^{3} + 8 \, {\left (12 \, {\left (B b^{2} c^{4} + 30 \, A b c^{5}\right )} d - {\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} e\right )} x^{2} + 180 \, {\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d - 15 \, {\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )} e - 10 \, {\left (12 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d - {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{7680 \, c^{5}}\right ] \] Input:

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

Output:

[1/15360*(15*(12*(B*b^5*c - 2*A*b^4*c^2)*d - (7*B*b^6 - 12*A*b^5*c)*e)*sqr 
t(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(1280*B*c^6*e*x^5 + 
128*(12*B*c^6*d + (13*B*b*c^5 + 12*A*c^6)*e)*x^4 + 48*(4*(11*B*b*c^5 + 10* 
A*c^6)*d + (B*b^2*c^4 + 44*A*b*c^5)*e)*x^3 + 8*(12*(B*b^2*c^4 + 30*A*b*c^5 
)*d - (7*B*b^3*c^3 - 12*A*b^2*c^4)*e)*x^2 + 180*(B*b^4*c^2 - 2*A*b^3*c^3)* 
d - 15*(7*B*b^5*c - 12*A*b^4*c^2)*e - 10*(12*(B*b^3*c^3 - 2*A*b^2*c^4)*d - 
 (7*B*b^4*c^2 - 12*A*b^3*c^3)*e)*x)*sqrt(c*x^2 + b*x))/c^5, 1/7680*(15*(12 
*(B*b^5*c - 2*A*b^4*c^2)*d - (7*B*b^6 - 12*A*b^5*c)*e)*sqrt(-c)*arctan(sqr 
t(c*x^2 + b*x)*sqrt(-c)/(c*x + b)) + (1280*B*c^6*e*x^5 + 128*(12*B*c^6*d + 
 (13*B*b*c^5 + 12*A*c^6)*e)*x^4 + 48*(4*(11*B*b*c^5 + 10*A*c^6)*d + (B*b^2 
*c^4 + 44*A*b*c^5)*e)*x^3 + 8*(12*(B*b^2*c^4 + 30*A*b*c^5)*d - (7*B*b^3*c^ 
3 - 12*A*b^2*c^4)*e)*x^2 + 180*(B*b^4*c^2 - 2*A*b^3*c^3)*d - 15*(7*B*b^5*c 
 - 12*A*b^4*c^2)*e - 10*(12*(B*b^3*c^3 - 2*A*b^2*c^4)*d - (7*B*b^4*c^2 - 1 
2*A*b^3*c^3)*e)*x)*sqrt(c*x^2 + b*x))/c^5]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (314) = 628\).

Time = 0.93 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.10 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 b^{2} \left (A b^{2} d - \frac {5 b \left (A b^{2} e + 2 A b c d + B b^{2} d - \frac {7 b \left (2 A b c e + A c^{2} d + B b^{2} e + 2 B b c d - \frac {9 b \left (A c^{2} e + \frac {13 B b c e}{12} + B c^{2} d\right )}{10 c}\right )}{8 c}\right )}{6 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{8 c^{2}} + \sqrt {b x + c x^{2}} \left (\frac {B c e x^{5}}{6} - \frac {3 b \left (A b^{2} d - \frac {5 b \left (A b^{2} e + 2 A b c d + B b^{2} d - \frac {7 b \left (2 A b c e + A c^{2} d + B b^{2} e + 2 B b c d - \frac {9 b \left (A c^{2} e + \frac {13 B b c e}{12} + B c^{2} d\right )}{10 c}\right )}{8 c}\right )}{6 c}\right )}{4 c^{2}} + \frac {x^{4} \left (A c^{2} e + \frac {13 B b c e}{12} + B c^{2} d\right )}{5 c} + \frac {x^{3} \cdot \left (2 A b c e + A c^{2} d + B b^{2} e + 2 B b c d - \frac {9 b \left (A c^{2} e + \frac {13 B b c e}{12} + B c^{2} d\right )}{10 c}\right )}{4 c} + \frac {x^{2} \left (A b^{2} e + 2 A b c d + B b^{2} d - \frac {7 b \left (2 A b c e + A c^{2} d + B b^{2} e + 2 B b c d - \frac {9 b \left (A c^{2} e + \frac {13 B b c e}{12} + B c^{2} d\right )}{10 c}\right )}{8 c}\right )}{3 c} + \frac {x \left (A b^{2} d - \frac {5 b \left (A b^{2} e + 2 A b c d + B b^{2} d - \frac {7 b \left (2 A b c e + A c^{2} d + B b^{2} e + 2 B b c d - \frac {9 b \left (A c^{2} e + \frac {13 B b c e}{12} + B c^{2} d\right )}{10 c}\right )}{8 c}\right )}{6 c}\right )}{2 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {A d \left (b x\right )^{\frac {5}{2}}}{5} + \frac {B e \left (b x\right )^{\frac {9}{2}}}{9 b^{2}} + \frac {\left (b x\right )^{\frac {7}{2}} \left (A e + B d\right )}{7 b}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((3*b**2*(A*b**2*d - 5*b*(A*b**2*e + 2*A*b*c*d + B*b**2*d - 7*b*( 
2*A*b*c*e + A*c**2*d + B*b**2*e + 2*B*b*c*d - 9*b*(A*c**2*e + 13*B*b*c*e/1 
2 + B*c**2*d)/(10*c))/(8*c))/(6*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x 
+ c*x**2) + 2*c*x)/sqrt(c), Ne(b**2/c, 0)), ((b/(2*c) + x)*log(b/(2*c) + x 
)/sqrt(c*(b/(2*c) + x)**2), True))/(8*c**2) + sqrt(b*x + c*x**2)*(B*c*e*x* 
*5/6 - 3*b*(A*b**2*d - 5*b*(A*b**2*e + 2*A*b*c*d + B*b**2*d - 7*b*(2*A*b*c 
*e + A*c**2*d + B*b**2*e + 2*B*b*c*d - 9*b*(A*c**2*e + 13*B*b*c*e/12 + B*c 
**2*d)/(10*c))/(8*c))/(6*c))/(4*c**2) + x**4*(A*c**2*e + 13*B*b*c*e/12 + B 
*c**2*d)/(5*c) + x**3*(2*A*b*c*e + A*c**2*d + B*b**2*e + 2*B*b*c*d - 9*b*( 
A*c**2*e + 13*B*b*c*e/12 + B*c**2*d)/(10*c))/(4*c) + x**2*(A*b**2*e + 2*A* 
b*c*d + B*b**2*d - 7*b*(2*A*b*c*e + A*c**2*d + B*b**2*e + 2*B*b*c*d - 9*b* 
(A*c**2*e + 13*B*b*c*e/12 + B*c**2*d)/(10*c))/(8*c))/(3*c) + x*(A*b**2*d - 
 5*b*(A*b**2*e + 2*A*b*c*d + B*b**2*d - 7*b*(2*A*b*c*e + A*c**2*d + B*b**2 
*e + 2*B*b*c*d - 9*b*(A*c**2*e + 13*B*b*c*e/12 + B*c**2*d)/(10*c))/(8*c))/ 
(6*c))/(2*c)), Ne(c, 0)), (2*(A*d*(b*x)**(5/2)/5 + B*e*(b*x)**(9/2)/(9*b** 
2) + (b*x)**(7/2)*(A*e + B*d)/(7*b))/b, Ne(b, 0)), (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.37 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A d x - \frac {3 \, \sqrt {c x^{2} + b x} A b^{2} d x}{32 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{4} e x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{2} e x}{96 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B e x}{6 \, c} + \frac {3 \, A b^{4} d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} + \frac {7 \, B b^{6} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{3} d}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b d}{8 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{5} e}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} e}{192 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b e}{60 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + b x} {\left (B d + A e\right )} b^{3} x}{64 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B d + A e\right )} b x}{8 \, c} - \frac {3 \, {\left (B d + A e\right )} b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} + \frac {3 \, \sqrt {c x^{2} + b x} {\left (B d + A e\right )} b^{4}}{128 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B d + A e\right )} b^{2}}{16 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B d + A e\right )}}{5 \, c} \] Input:

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

Output:

1/4*(c*x^2 + b*x)^(3/2)*A*d*x - 3/32*sqrt(c*x^2 + b*x)*A*b^2*d*x/c - 7/256 
*sqrt(c*x^2 + b*x)*B*b^4*e*x/c^3 + 7/96*(c*x^2 + b*x)^(3/2)*B*b^2*e*x/c^2 
+ 1/6*(c*x^2 + b*x)^(5/2)*B*e*x/c + 3/128*A*b^4*d*log(2*c*x + b + 2*sqrt(c 
*x^2 + b*x)*sqrt(c))/c^(5/2) + 7/1024*B*b^6*e*log(2*c*x + b + 2*sqrt(c*x^2 
 + b*x)*sqrt(c))/c^(9/2) - 3/64*sqrt(c*x^2 + b*x)*A*b^3*d/c^2 + 1/8*(c*x^2 
 + b*x)^(3/2)*A*b*d/c - 7/512*sqrt(c*x^2 + b*x)*B*b^5*e/c^4 + 7/192*(c*x^2 
 + b*x)^(3/2)*B*b^3*e/c^3 - 7/60*(c*x^2 + b*x)^(5/2)*B*b*e/c^2 + 3/64*sqrt 
(c*x^2 + b*x)*(B*d + A*e)*b^3*x/c^2 - 1/8*(c*x^2 + b*x)^(3/2)*(B*d + A*e)* 
b*x/c - 3/256*(B*d + A*e)*b^5*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) 
/c^(7/2) + 3/128*sqrt(c*x^2 + b*x)*(B*d + A*e)*b^4/c^3 - 1/16*(c*x^2 + b*x 
)^(3/2)*(B*d + A*e)*b^2/c^2 + 1/5*(c*x^2 + b*x)^(5/2)*(B*d + A*e)/c
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.95 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B c e x + \frac {12 \, B c^{6} d + 13 \, B b c^{5} e + 12 \, A c^{6} e}{c^{5}}\right )} x + \frac {3 \, {\left (44 \, B b c^{5} d + 40 \, A c^{6} d + B b^{2} c^{4} e + 44 \, A b c^{5} e\right )}}{c^{5}}\right )} x + \frac {12 \, B b^{2} c^{4} d + 360 \, A b c^{5} d - 7 \, B b^{3} c^{3} e + 12 \, A b^{2} c^{4} e}{c^{5}}\right )} x - \frac {5 \, {\left (12 \, B b^{3} c^{3} d - 24 \, A b^{2} c^{4} d - 7 \, B b^{4} c^{2} e + 12 \, A b^{3} c^{3} e\right )}}{c^{5}}\right )} x + \frac {15 \, {\left (12 \, B b^{4} c^{2} d - 24 \, A b^{3} c^{3} d - 7 \, B b^{5} c e + 12 \, A b^{4} c^{2} e\right )}}{c^{5}}\right )} + \frac {{\left (12 \, B b^{5} c d - 24 \, A b^{4} c^{2} d - 7 \, B b^{6} e + 12 \, A b^{5} c e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \] Input:

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

Output:

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*B*c*e*x + (12*B*c^6*d + 13*B*b*c^ 
5*e + 12*A*c^6*e)/c^5)*x + 3*(44*B*b*c^5*d + 40*A*c^6*d + B*b^2*c^4*e + 44 
*A*b*c^5*e)/c^5)*x + (12*B*b^2*c^4*d + 360*A*b*c^5*d - 7*B*b^3*c^3*e + 12* 
A*b^2*c^4*e)/c^5)*x - 5*(12*B*b^3*c^3*d - 24*A*b^2*c^4*d - 7*B*b^4*c^2*e + 
 12*A*b^3*c^3*e)/c^5)*x + 15*(12*B*b^4*c^2*d - 24*A*b^3*c^3*d - 7*B*b^5*c* 
e + 12*A*b^4*c^2*e)/c^5) + 1/1024*(12*B*b^5*c*d - 24*A*b^4*c^2*d - 7*B*b^6 
*e + 12*A*b^5*c*e)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b)) 
/c^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )\,\left (d+e\,x\right ) \,d x \] 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 = 0.31 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.55 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {180 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{4} c^{2} e -360 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{3} c^{3} d -120 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{3} c^{3} e x +240 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{2} c^{4} d x +96 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{2} c^{4} e \,x^{2}+2880 \sqrt {x}\, \sqrt {c x +b}\, a b \,c^{5} d \,x^{2}+2112 \sqrt {x}\, \sqrt {c x +b}\, a b \,c^{5} e \,x^{3}+1920 \sqrt {x}\, \sqrt {c x +b}\, a \,c^{6} d \,x^{3}+1536 \sqrt {x}\, \sqrt {c x +b}\, a \,c^{6} e \,x^{4}-105 \sqrt {x}\, \sqrt {c x +b}\, b^{6} c e +180 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c^{2} d +70 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c^{2} e x -120 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{3} d x -56 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{3} e \,x^{2}+96 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{4} d \,x^{2}+48 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{4} e \,x^{3}+2112 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{5} d \,x^{3}+1664 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{5} e \,x^{4}+1536 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{6} d \,x^{4}+1280 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{6} e \,x^{5}-180 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) a \,b^{5} c e +360 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) a \,b^{4} c^{2} d +105 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{7} e -180 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{6} c d}{7680 c^{5}} \] Input:

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

Output:

(180*sqrt(x)*sqrt(b + c*x)*a*b**4*c**2*e - 360*sqrt(x)*sqrt(b + c*x)*a*b** 
3*c**3*d - 120*sqrt(x)*sqrt(b + c*x)*a*b**3*c**3*e*x + 240*sqrt(x)*sqrt(b 
+ c*x)*a*b**2*c**4*d*x + 96*sqrt(x)*sqrt(b + c*x)*a*b**2*c**4*e*x**2 + 288 
0*sqrt(x)*sqrt(b + c*x)*a*b*c**5*d*x**2 + 2112*sqrt(x)*sqrt(b + c*x)*a*b*c 
**5*e*x**3 + 1920*sqrt(x)*sqrt(b + c*x)*a*c**6*d*x**3 + 1536*sqrt(x)*sqrt( 
b + c*x)*a*c**6*e*x**4 - 105*sqrt(x)*sqrt(b + c*x)*b**6*c*e + 180*sqrt(x)* 
sqrt(b + c*x)*b**5*c**2*d + 70*sqrt(x)*sqrt(b + c*x)*b**5*c**2*e*x - 120*s 
qrt(x)*sqrt(b + c*x)*b**4*c**3*d*x - 56*sqrt(x)*sqrt(b + c*x)*b**4*c**3*e* 
x**2 + 96*sqrt(x)*sqrt(b + c*x)*b**3*c**4*d*x**2 + 48*sqrt(x)*sqrt(b + c*x 
)*b**3*c**4*e*x**3 + 2112*sqrt(x)*sqrt(b + c*x)*b**2*c**5*d*x**3 + 1664*sq 
rt(x)*sqrt(b + c*x)*b**2*c**5*e*x**4 + 1536*sqrt(x)*sqrt(b + c*x)*b*c**6*d 
*x**4 + 1280*sqrt(x)*sqrt(b + c*x)*b*c**6*e*x**5 - 180*sqrt(c)*log((sqrt(b 
 + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*a*b**5*c*e + 360*sqrt(c)*log((sqrt(b + 
 c*x) + sqrt(x)*sqrt(c))/sqrt(b))*a*b**4*c**2*d + 105*sqrt(c)*log((sqrt(b 
+ c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**7*e - 180*sqrt(c)*log((sqrt(b + c*x) 
 + sqrt(x)*sqrt(c))/sqrt(b))*b**6*c*d)/(7680*c**5)