\(\int \sqrt {e x} (a+b x^3)^{3/2} (A+B x^3) \, dx\) [252]

Optimal result

Integrand size = 26, antiderivative size = 161 \[ \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {a (6 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 b e}+\frac {(6 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}+\frac {a^2 (6 A b-a B) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 b^{3/2}} \] Output:

1/24*a*(6*A*b-B*a)*(e*x)^(3/2)*(b*x^3+a)^(1/2)/b/e+1/36*(6*A*b-B*a)*(e*x)^ 
(3/2)*(b*x^3+a)^(3/2)/b/e+1/9*B*(e*x)^(3/2)*(b*x^3+a)^(5/2)/b/e+1/24*a^2*( 
6*A*b-B*a)*e^(1/2)*arctanh(b^(1/2)*(e*x)^(3/2)/e^(3/2)/(b*x^3+a)^(1/2))/b^ 
(3/2)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.75 \[ \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {x \sqrt {e x} \sqrt {a+b x^3} \left (30 a A b+3 a^2 B+12 A b^2 x^3+14 a b B x^3+8 b^2 B x^6\right )}{72 b}-\frac {a^2 (-6 A b+a B) \sqrt {e x} \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )}{24 b^{3/2} \sqrt {x}} \] Input:

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

Output:

(x*Sqrt[e*x]*Sqrt[a + b*x^3]*(30*a*A*b + 3*a^2*B + 12*A*b^2*x^3 + 14*a*b*B 
*x^3 + 8*b^2*B*x^6))/(72*b) - (a^2*(-6*A*b + a*B)*Sqrt[e*x]*Log[Sqrt[b]*x^ 
(3/2) + Sqrt[a + b*x^3]])/(24*b^(3/2)*Sqrt[x])
 

Rubi [A] (warning: unable to verify)

Time = 0.51 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {959, 811, 811, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 811

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

\(\Big \downarrow \) 811

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

\(\Big \downarrow \) 851

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(B*(e*x)^(3/2)*(a + b*x^3)^(5/2))/(9*b*e) + ((6*A*b - a*B)*(((e*x)^(3/2)*( 
a + b*x^3)^(3/2))/(6*e) + (3*a*(((e*x)^(3/2)*Sqrt[a + b*x^3])/(3*e) + (a*S 
qrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + (b*x)/e^2])])/(3*Sq 
rt[b])))/4))/(6*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 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 = 1.62 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.83

method	result	size
risch	\(\frac {x^{2} \left (8 b^{2} B \,x^{6}+12 A \,b^{2} x^{3}+14 B a b \,x^{3}+30 a b A +3 a^{2} B \right ) \sqrt {b \,x^{3}+a}\, e}{72 b \sqrt {e x}}+\frac {a^{2} \left (6 A b -B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {\left (b \,x^{3}+a \right ) e x}}{x^{2} \sqrt {b e}}\right ) e \sqrt {\left (b \,x^{3}+a \right ) e x}}{24 b \sqrt {b e}\, \sqrt {e x}\, \sqrt {b \,x^{3}+a}}\)	\(134\)
default	\(\frac {\sqrt {e x}\, \sqrt {b \,x^{3}+a}\, \left (8 B \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, b^{2} x^{7}+12 A \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, b^{2} x^{4}+14 B \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, a b \,x^{4}+30 A \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, a b x +18 A \,\operatorname {arctanh}\left (\frac {\sqrt {\left (b \,x^{3}+a \right ) e x}}{x^{2} \sqrt {b e}}\right ) a^{2} b e +3 B \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, a^{2} x -3 B \,\operatorname {arctanh}\left (\frac {\sqrt {\left (b \,x^{3}+a \right ) e x}}{x^{2} \sqrt {b e}}\right ) a^{3} e \right )}{72 \sqrt {\left (b \,x^{3}+a \right ) e x}\, \sqrt {b e}\, b}\)	\(221\)
elliptic	\(\text {Expression too large to display}\)	\(1183\)

Input:

int((e*x)^(1/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x,method=_RETURNVERBOSE)
 

Output:

1/72/b*x^2*(8*B*b^2*x^6+12*A*b^2*x^3+14*B*a*b*x^3+30*A*a*b+3*B*a^2)*(b*x^3 
+a)^(1/2)*e/(e*x)^(1/2)+1/24*a^2/b*(6*A*b-B*a)/(b*e)^(1/2)*arctanh(((b*x^3 
+a)*e*x)^(1/2)/x^2/(b*e)^(1/2))*e*((b*x^3+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^3 
+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.70 \[ \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\left [-\frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {\frac {e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {e x} \sqrt {\frac {e}{b}}\right ) - 4 \, {\left (8 \, B b^{2} x^{7} + 2 \, {\left (7 \, B a b + 6 \, A b^{2}\right )} x^{4} + 3 \, {\left (B a^{2} + 10 \, A a b\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{288 \, b}, \frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {e x} b x \sqrt {-\frac {e}{b}}}{2 \, b e x^{3} + a e}\right ) + 2 \, {\left (8 \, B b^{2} x^{7} + 2 \, {\left (7 \, B a b + 6 \, A b^{2}\right )} x^{4} + 3 \, {\left (B a^{2} + 10 \, A a b\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{144 \, b}\right ] \] Input:

integrate((e*x)^(1/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="fricas")
 

Output:

[-1/288*(3*(B*a^3 - 6*A*a^2*b)*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*(8* 
B*b^2*x^7 + 2*(7*B*a*b + 6*A*b^2)*x^4 + 3*(B*a^2 + 10*A*a*b)*x)*sqrt(b*x^3 
 + a)*sqrt(e*x))/b, 1/144*(3*(B*a^3 - 6*A*a^2*b)*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*(8*B*b^2*x^7 + 
2*(7*B*a*b + 6*A*b^2)*x^4 + 3*(B*a^2 + 10*A*a*b)*x)*sqrt(b*x^3 + a)*sqrt(e 
*x))/b]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (138) = 276\).

Time = 3.81 (sec) , antiderivative size = 546, normalized size of antiderivative = 3.39 \[ \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx =\text {Too large to display} \] Input:

integrate((e*x)**(1/2)*(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*Piecewi 
se((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))/2 + ( 
e*x)**(3/2)*sqrt(a + b*x**3)/2, Ne(b/e**3, 0)), (sqrt(a)*(e*x)**(3/2), Tru 
e)) + A*b*Piecewise((-a**2*e**3*Piecewise((log(2*b*(e*x)**(3/2)/e**3 + 2*s 
qrt(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))/(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, Tr 
ue)) + B*a*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), True))/(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, T 
rue)) + B*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), 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))/e**3)/(3*e**3), True))/e, 
 Ne(e, 0)), (0, True))
 

Maxima [F]

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

integrate((e*x)^(1/2)*(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)*sqrt(e*x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (126) = 252\).

Time = 0.25 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.58 \[ \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {\sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, e^{3} x^{3} {\left (\frac {4 \, x^{3}}{e^{4}} + \frac {a}{b e^{4}}\right )} - \frac {3 \, a^{2}}{b^{2} e}\right )} \sqrt {e x} B b x {\left | e \right |}^{2}}{72 \, e^{3}} - \frac {{\left (B^{2} a^{6} + 4 \, A B a^{5} b + 4 \, A^{2} a^{4} b^{2}\right )} e^{4} \log \left ({\left | {\left (\sqrt {e x} B a^{3} x + 2 \, \sqrt {e x} A a^{2} b x\right )} \sqrt {b e} + \sqrt {B^{2} a^{7} e^{2} + 4 \, A B a^{6} b e^{2} + 4 \, A^{2} a^{5} b^{2} e^{2} + {\left (\sqrt {e x} B a^{3} x + 2 \, \sqrt {e x} A a^{2} b x\right )}^{2} b e} \right |}\right )}{24 \, \sqrt {b e} b {\left | B a^{3} e + 2 \, A a^{2} b e \right |} {\left | e \right |}^{2}} - \frac {{\left (\frac {a e^{4} \log \left ({\left | -\sqrt {b e} \sqrt {e x} e x + \sqrt {b e^{4} x^{3} + a e^{4}} \right |}\right )}{\sqrt {b e}} - \sqrt {b e^{4} x^{3} + a e^{4}} \sqrt {e x} e x\right )} A a {\left | e \right |}^{2}}{3 \, e^{5}} + \frac {\sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, e^{3} x^{3} + \frac {a e^{3}}{b}\right )} \sqrt {e x} B a x {\left | e \right |}^{2}}{12 \, e^{7}} + \frac {\sqrt {b e^{4} x^{3} + a e^{4}} {\left (2 \, e^{3} x^{3} + \frac {a e^{3}}{b}\right )} \sqrt {e x} A b x {\left | e \right |}^{2}}{12 \, e^{7}} \] Input:

integrate((e*x)^(1/2)*(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*b*x*abs(e)^2/e^3 - 1/24*(B^2*a^6 + 4*A*B*a^5*b + 4*A^2* 
a^4*b^2)*e^4*log(abs((sqrt(e*x)*B*a^3*x + 2*sqrt(e*x)*A*a^2*b*x)*sqrt(b*e) 
 + sqrt(B^2*a^7*e^2 + 4*A*B*a^6*b*e^2 + 4*A^2*a^5*b^2*e^2 + (sqrt(e*x)*B*a 
^3*x + 2*sqrt(e*x)*A*a^2*b*x)^2*b*e)))/(sqrt(b*e)*b*abs(B*a^3*e + 2*A*a^2* 
b*e)*abs(e)^2) - 1/3*(a*e^4*log(abs(-sqrt(b*e)*sqrt(e*x)*e*x + sqrt(b*e^4* 
x^3 + a*e^4)))/sqrt(b*e) - sqrt(b*e^4*x^3 + a*e^4)*sqrt(e*x)*e*x)*A*a*abs( 
e)^2/e^5 + 1/12*sqrt(b*e^4*x^3 + a*e^4)*(2*e^3*x^3 + a*e^3/b)*sqrt(e*x)*B* 
a*x*abs(e)^2/e^7 + 1/12*sqrt(b*e^4*x^3 + a*e^4)*(2*e^3*x^3 + a*e^3/b)*sqrt 
(e*x)*A*b*x*abs(e)^2/e^7
 

Mupad [F(-1)]

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

int((A + B*x^3)*(e*x)^(1/2)*(a + b*x^3)^(3/2),x)
 

Output:

int((A + B*x^3)*(e*x)^(1/2)*(a + b*x^3)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.68 \[ \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {\sqrt {e}\, \left (66 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a^{2} b x +52 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a \,b^{2} x^{4}+16 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b^{3} x^{7}-15 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, x \right ) a^{3}+15 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {x}\, \sqrt {b}\, x \right ) a^{3}\right )}{144 b} \] Input:

int((e*x)^(1/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x)
 

Output:

(sqrt(e)*(66*sqrt(x)*sqrt(a + b*x**3)*a**2*b*x + 52*sqrt(x)*sqrt(a + b*x** 
3)*a*b**2*x**4 + 16*sqrt(x)*sqrt(a + b*x**3)*b**3*x**7 - 15*sqrt(b)*log(sq 
rt(a + b*x**3) - sqrt(x)*sqrt(b)*x)*a**3 + 15*sqrt(b)*log(sqrt(a + b*x**3) 
 + sqrt(x)*sqrt(b)*x)*a**3))/(144*b)