\(\int (e x)^{7/2} (a+b x^3)^{3/2} (A+B x^3) \, dx\) [249]

Optimal result

Integrand size = 26, antiderivative size = 201 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {a^3 (8 A b-3 a B) e^{7/2} \text {arctanh}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{5/2}} \] Output:

1/192*a^2*(8*A*b-3*B*a)*e^2*(e*x)^(3/2)*(b*x^3+a)^(1/2)/b^2+1/96*a*(8*A*b- 
3*B*a)*(e*x)^(9/2)*(b*x^3+a)^(1/2)/b/e+1/72*(8*A*b-3*B*a)*(e*x)^(9/2)*(b*x 
^3+a)^(3/2)/b/e+1/12*B*(e*x)^(9/2)*(b*x^3+a)^(5/2)/b/e-1/192*a^3*(8*A*b-3* 
B*a)*e^(7/2)*arctanh(b^(1/2)*(e*x)^(3/2)/e^(3/2)/(b*x^3+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {e^3 \sqrt {e x} \left (\sqrt {b} x^{3/2} \sqrt {a+b x^3} \left (-9 a^3 B+6 a^2 b \left (4 A+B x^3\right )+16 b^3 x^6 \left (4 A+3 B x^3\right )+8 a b^2 x^3 \left (14 A+9 B x^3\right )\right )+3 a^3 (-8 A b+3 a B) \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )\right )}{576 b^{5/2} \sqrt {x}} \] Input:

Integrate[(e*x)^(7/2)*(a + b*x^3)^(3/2)*(A + B*x^3),x]
 

Output:

(e^3*Sqrt[e*x]*(Sqrt[b]*x^(3/2)*Sqrt[a + b*x^3]*(-9*a^3*B + 6*a^2*b*(4*A + 
 B*x^3) + 16*b^3*x^6*(4*A + 3*B*x^3) + 8*a*b^2*x^3*(14*A + 9*B*x^3)) + 3*a 
^3*(-8*A*b + 3*a*B)*Log[Sqrt[b]*x^(3/2) + Sqrt[a + b*x^3]]))/(576*b^(5/2)* 
Sqrt[x])
 

Rubi [A] (warning: unable to verify)

Time = 0.59 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {959, 811, 811, 843, 851, 807, 224, 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.

\(\displaystyle \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {(8 A b-3 a B) \int (e x)^{7/2} \left (b x^3+a\right )^{3/2}dx}{8 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {(8 A b-3 a B) \left (\frac {1}{2} a \int (e x)^{7/2} \sqrt {b x^3+a}dx+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )}{8 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {(8 A b-3 a B) \left (\frac {1}{2} a \left (\frac {1}{4} a \int \frac {(e x)^{7/2}}{\sqrt {b x^3+a}}dx+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )}{8 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {(8 A b-3 a B) \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^3 \int \frac {\sqrt {e x}}{\sqrt {b x^3+a}}dx}{2 b}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )}{8 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {(8 A b-3 a B) \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^2 \int \frac {e x}{\sqrt {b x^3+a}}d\sqrt {e x}}{b}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )}{8 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {(8 A b-3 a B) \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^2 \int \frac {1}{\sqrt {a+\frac {b x}{e^2}}}d(e x)^{3/2}}{3 b}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )}{8 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {(8 A b-3 a B) \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^2 \int \frac {1}{1-\frac {b x}{e^2}}d\frac {(e x)^{3/2}}{\sqrt {a+\frac {b x}{e^2}}}}{3 b}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )}{8 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(8 A b-3 a B) \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {e^2 (e x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a e^{7/2} \text {arctanh}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+\frac {b x}{e^2}}}\right )}{3 b^{3/2}}\right )+\frac {(e x)^{9/2} \sqrt {a+b x^3}}{6 e}\right )+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 e}\right )}{8 b}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}\)

Input:

Int[(e*x)^(7/2)*(a + b*x^3)^(3/2)*(A + B*x^3),x]
 

Output:

(B*(e*x)^(9/2)*(a + b*x^3)^(5/2))/(12*b*e) + ((8*A*b - 3*a*B)*(((e*x)^(9/2 
)*(a + b*x^3)^(3/2))/(9*e) + (a*(((e*x)^(9/2)*Sqrt[a + b*x^3])/(6*e) + (a* 
((e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(3*b) - (a*e^(7/2)*ArcTanh[(Sqrt[b]*(e* 
x)^(3/2))/(e^(3/2)*Sqrt[a + (b*x)/e^2])])/(3*b^(3/2))))/4))/2))/(8*b)
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 811

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 
))   Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I 
GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m 
, p, x]
 

rule 843

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n 
 - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ 
a*c^n*((m - n + 1)/(b*(m + n*p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^p, x] 
, x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* 
p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

Maple [A] (verified)

Time = 2.90 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.81

method	result	size
risch	\(\frac {x^{2} \left (48 b^{3} B \,x^{9}+64 A \,b^{3} x^{6}+72 B a \,b^{2} x^{6}+112 a A \,b^{2} x^{3}+6 B \,a^{2} b \,x^{3}+24 a^{2} b A -9 a^{3} B \right ) \sqrt {b \,x^{3}+a}\, e^{4}}{576 b^{2} \sqrt {e x}}-\frac {a^{3} \left (8 A b -3 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {\left (b \,x^{3}+a \right ) e x}}{x^{2} \sqrt {b e}}\right ) e^{4} \sqrt {\left (b \,x^{3}+a \right ) e x}}{192 b^{2} \sqrt {b e}\, \sqrt {e x}\, \sqrt {b \,x^{3}+a}}\)	\(162\)
default	\(-\frac {e^{3} \sqrt {e x}\, \sqrt {b \,x^{3}+a}\, \left (-48 B \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, b^{3} x^{10}-64 A \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, b^{3} x^{7}-72 B \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, a \,b^{2} x^{7}-112 A \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, a \,b^{2} x^{4}-6 B \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, a^{2} b \,x^{4}+24 A \,\operatorname {arctanh}\left (\frac {\sqrt {\left (b \,x^{3}+a \right ) e x}}{x^{2} \sqrt {b e}}\right ) a^{3} b e -24 A \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, a^{2} b x -9 B \,\operatorname {arctanh}\left (\frac {\sqrt {\left (b \,x^{3}+a \right ) e x}}{x^{2} \sqrt {b e}}\right ) a^{4} e +9 B \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, a^{3} x \right )}{576 \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, b^{2}}\)	\(282\)
elliptic	\(\text {Expression too large to display}\)	\(1285\)

Input:

int((e*x)^(7/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x,method=_RETURNVERBOSE)
 

Output:

1/576/b^2*x^2*(48*B*b^3*x^9+64*A*b^3*x^6+72*B*a*b^2*x^6+112*A*a*b^2*x^3+6* 
B*a^2*b*x^3+24*A*a^2*b-9*B*a^3)*(b*x^3+a)^(1/2)*e^4/(e*x)^(1/2)-1/192*a^3/ 
b^2*(8*A*b-3*B*a)/(b*e)^(1/2)*arctanh(((b*x^3+a)*e*x)^(1/2)/x^2/(b*e)^(1/2 
))*e^4*((b*x^3+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^3+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.77 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\left [-\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{3} \sqrt {\frac {e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {e x} \sqrt {\frac {e}{b}}\right ) - 4 \, {\left (48 \, B b^{3} e^{3} x^{10} + 8 \, {\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} e^{3} x^{7} + 2 \, {\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} e^{3} x^{4} - 3 \, {\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{2304 \, b^{2}}, -\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{3} \sqrt {-\frac {e}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {e x} b x \sqrt {-\frac {e}{b}}}{2 \, b e x^{3} + a e}\right ) - 2 \, {\left (48 \, B b^{3} e^{3} x^{10} + 8 \, {\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} e^{3} x^{7} + 2 \, {\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} e^{3} x^{4} - 3 \, {\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{1152 \, b^{2}}\right ] \] Input:

integrate((e*x)^(7/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="fricas")
 

Output:

[-1/2304*(3*(3*B*a^4 - 8*A*a^3*b)*e^3*sqrt(e/b)*log(-8*b^2*e*x^6 - 8*a*b*e 
*x^3 - a^2*e + 4*(2*b^2*x^4 + a*b*x)*sqrt(b*x^3 + a)*sqrt(e*x)*sqrt(e/b)) 
- 4*(48*B*b^3*e^3*x^10 + 8*(9*B*a*b^2 + 8*A*b^3)*e^3*x^7 + 2*(3*B*a^2*b + 
56*A*a*b^2)*e^3*x^4 - 3*(3*B*a^3 - 8*A*a^2*b)*e^3*x)*sqrt(b*x^3 + a)*sqrt( 
e*x))/b^2, -1/1152*(3*(3*B*a^4 - 8*A*a^3*b)*e^3*sqrt(-e/b)*arctan(2*sqrt(b 
*x^3 + a)*sqrt(e*x)*b*x*sqrt(-e/b)/(2*b*e*x^3 + a*e)) - 2*(48*B*b^3*e^3*x^ 
10 + 8*(9*B*a*b^2 + 8*A*b^3)*e^3*x^7 + 2*(3*B*a^2*b + 56*A*a*b^2)*e^3*x^4 
- 3*(3*B*a^3 - 8*A*a^2*b)*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b^2]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (180) = 360\).

Time = 33.11 (sec) , antiderivative size = 634, normalized size of antiderivative = 3.15 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((e*x)**(7/2)*(b*x**3+a)**(3/2)*(B*x**3+A),x)
 

Output:

Piecewise((2*Piecewise((nan, Eq(e**3, 0)), ((A*a*e**3*Piecewise((-a**2*e** 
3*Piecewise((log(2*b*(e*x)**(3/2)/e**3 + 2*sqrt(b/e**3)*sqrt(a + b*x**3))/ 
sqrt(b/e**3), Ne(a, 0)), ((e*x)**(3/2)*log((e*x)**(3/2))/sqrt(b*x**3), Tru 
e))/(8*b) + sqrt(a + b*x**3)*(a*e**3*(e*x)**(3/2)/(8*b) + (e*x)**(9/2)/4), 
 Ne(b/e**3, 0)), (sqrt(a)*(e*x)**(9/2)/3, True)) + A*b*Piecewise((a**3*e** 
6*Piecewise((log(2*b*(e*x)**(3/2)/e**3 + 2*sqrt(b/e**3)*sqrt(a + b*x**3))/ 
sqrt(b/e**3), Ne(a, 0)), ((e*x)**(3/2)*log((e*x)**(3/2))/sqrt(b*x**3), Tru 
e))/(16*b**2) + sqrt(a + b*x**3)*(-a**2*e**6*(e*x)**(3/2)/(16*b**2) + a*e* 
*3*(e*x)**(9/2)/(24*b) + (e*x)**(15/2)/6), Ne(b/e**3, 0)), (sqrt(a)*(e*x)* 
*(15/2)/5, True)) + B*a*Piecewise((a**3*e**6*Piecewise((log(2*b*(e*x)**(3/ 
2)/e**3 + 2*sqrt(b/e**3)*sqrt(a + b*x**3))/sqrt(b/e**3), Ne(a, 0)), ((e*x) 
**(3/2)*log((e*x)**(3/2))/sqrt(b*x**3), True))/(16*b**2) + sqrt(a + b*x**3 
)*(-a**2*e**6*(e*x)**(3/2)/(16*b**2) + a*e**3*(e*x)**(9/2)/(24*b) + (e*x)* 
*(15/2)/6), Ne(b/e**3, 0)), (sqrt(a)*(e*x)**(15/2)/5, True)) + B*b*Piecewi 
se((-5*a**4*e**9*Piecewise((log(2*b*(e*x)**(3/2)/e**3 + 2*sqrt(b/e**3)*sqr 
t(a + b*x**3))/sqrt(b/e**3), Ne(a, 0)), ((e*x)**(3/2)*log((e*x)**(3/2))/sq 
rt(b*x**3), True))/(128*b**3) + sqrt(a + b*x**3)*(5*a**3*e**9*(e*x)**(3/2) 
/(128*b**3) - 5*a**2*e**6*(e*x)**(9/2)/(192*b**2) + a*e**3*(e*x)**(15/2)/( 
48*b) + (e*x)**(21/2)/8), Ne(b/e**3, 0)), (sqrt(a)*(e*x)**(21/2)/7, True)) 
/e**3)/(3*e**3), True))/e, Ne(e, 0)), (0, True))
 

Maxima [F]

\[ \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}} \,d x } \] Input:

integrate((e*x)^(7/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="maxima")
 

Output:

integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*(e*x)^(7/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (163) = 326\).

Time = 0.28 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.46 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {1}{72} \, \sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, e^{3} x^{3} {\left (\frac {4 \, x^{3}}{e^{4}} + \frac {a}{b e^{4}}\right )} - \frac {3 \, a^{2}}{b^{2} e}\right )} \sqrt {e x} B a x {\left | e \right |}^{2} + \frac {1}{72} \, \sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, e^{3} x^{3} {\left (\frac {4 \, x^{3}}{e^{4}} + \frac {a}{b e^{4}}\right )} - \frac {3 \, a^{2}}{b^{2} e}\right )} \sqrt {e x} A b x {\left | e \right |}^{2} + \frac {\sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, e^{3} x^{3} + \frac {a e^{3}}{b}\right )} \sqrt {e x} A a x {\left | e \right |}^{2}}{12 \, e^{4}} - \frac {{\left (9 \, B^{2} a^{8} e - 48 \, A B a^{7} b e + 64 \, A^{2} a^{6} b^{2} e\right )}^{2} e^{5} \log \left ({\left | -{\left (3 \, \sqrt {e x} B a^{4} e^{2} x - 8 \, \sqrt {e x} A a^{3} b e^{2} x\right )} \sqrt {b e} + \sqrt {9 \, B^{2} a^{9} e^{6} - 48 \, A B a^{8} b e^{6} + 64 \, A^{2} a^{7} b^{2} e^{6} + {\left (3 \, \sqrt {e x} B a^{4} e^{2} x - 8 \, \sqrt {e x} A a^{3} b e^{2} x\right )}^{2} b e} \right |}\right )}{192 \, \sqrt {b e} b^{2} {\left | 9 \, B^{2} a^{8} e - 48 \, A B a^{7} b e + 64 \, A^{2} a^{6} b^{2} e \right |} {\left | -3 \, B a^{4} + 8 \, A a^{3} b \right |} {\left | e \right |}^{2}} + \frac {{\left (\frac {15 \, a^{3} e^{9}}{b^{3}} + 2 \, {\left (4 \, {\left (6 \, e^{3} x^{3} + \frac {a e^{3}}{b}\right )} e^{3} x^{3} - \frac {5 \, a^{2} e^{6}}{b^{2}}\right )} e^{3} x^{3}\right )} \sqrt {b e^{4} x^{3} + a e^{4}} \sqrt {e x} B b x {\left | e \right |}^{2}}{576 \, e^{10}} \] Input:

integrate((e*x)^(7/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="giac")
 

Output:

1/72*sqrt(b*e^4*x^3 + a*e^4)*(2*e^3*x^3*(4*x^3/e^4 + a/(b*e^4)) - 3*a^2/(b 
^2*e))*sqrt(e*x)*B*a*x*abs(e)^2 + 1/72*sqrt(b*e^4*x^3 + a*e^4)*(2*e^3*x^3* 
(4*x^3/e^4 + a/(b*e^4)) - 3*a^2/(b^2*e))*sqrt(e*x)*A*b*x*abs(e)^2 + 1/12*s 
qrt(b*e^4*x^3 + a*e^4)*(2*e^3*x^3 + a*e^3/b)*sqrt(e*x)*A*a*x*abs(e)^2/e^4 
- 1/192*(9*B^2*a^8*e - 48*A*B*a^7*b*e + 64*A^2*a^6*b^2*e)^2*e^5*log(abs(-( 
3*sqrt(e*x)*B*a^4*e^2*x - 8*sqrt(e*x)*A*a^3*b*e^2*x)*sqrt(b*e) + sqrt(9*B^ 
2*a^9*e^6 - 48*A*B*a^8*b*e^6 + 64*A^2*a^7*b^2*e^6 + (3*sqrt(e*x)*B*a^4*e^2 
*x - 8*sqrt(e*x)*A*a^3*b*e^2*x)^2*b*e)))/(sqrt(b*e)*b^2*abs(9*B^2*a^8*e - 
48*A*B*a^7*b*e + 64*A^2*a^6*b^2*e)*abs(-3*B*a^4 + 8*A*a^3*b)*abs(e)^2) + 1 
/576*(15*a^3*e^9/b^3 + 2*(4*(6*e^3*x^3 + a*e^3/b)*e^3*x^3 - 5*a^2*e^6/b^2) 
*e^3*x^3)*sqrt(b*e^4*x^3 + a*e^4)*sqrt(e*x)*B*b*x*abs(e)^2/e^10
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^{7/2}\,{\left (b\,x^3+a\right )}^{3/2} \,d x \] Input:

int((A + B*x^3)*(e*x)^(7/2)*(a + b*x^3)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int((A + B*x^3)*(e*x)^(7/2)*(a + b*x^3)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.66 \[ \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {\sqrt {e}\, e^{3} \left (30 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a^{3} b x +236 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a^{2} b^{2} x^{4}+272 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a \,b^{3} x^{7}+96 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b^{4} x^{10}+15 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, x \right ) a^{4}-15 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {x}\, \sqrt {b}\, x \right ) a^{4}\right )}{1152 b^{2}} \] Input:

int((e*x)^(7/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x)
 

Output:

(sqrt(e)*e**3*(30*sqrt(x)*sqrt(a + b*x**3)*a**3*b*x + 236*sqrt(x)*sqrt(a + 
 b*x**3)*a**2*b**2*x**4 + 272*sqrt(x)*sqrt(a + b*x**3)*a*b**3*x**7 + 96*sq 
rt(x)*sqrt(a + b*x**3)*b**4*x**10 + 15*sqrt(b)*log(sqrt(a + b*x**3) - sqrt 
(x)*sqrt(b)*x)*a**4 - 15*sqrt(b)*log(sqrt(a + b*x**3) + sqrt(x)*sqrt(b)*x) 
*a**4))/(1152*b**2)