\(\int x (a x+b x^3)^{3/2} \, dx\) [54]

Optimal result

Integrand size = 15, antiderivative size = 304 \[ \int x \left (a x+b x^3\right )^{3/2} \, dx=-\frac {8 a^3 x \left (a+b x^2\right )}{65 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {8 a^2 x \sqrt {a x+b x^3}}{195 b}+\frac {4}{39} a x^3 \sqrt {a x+b x^3}+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}+\frac {8 a^{13/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a x+b x^3}}-\frac {4 a^{13/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{65 b^{7/4} \sqrt {a x+b x^3}} \] Output:

-8/65*a^3*x*(b*x^2+a)/b^(3/2)/(a^(1/2)+b^(1/2)*x)/(b*x^3+a*x)^(1/2)+8/195* 
a^2*x*(b*x^3+a*x)^(1/2)/b+4/39*a*x^3*(b*x^3+a*x)^(1/2)+2/13*x^2*(b*x^3+a*x 
)^(3/2)+8/65*a^(13/4)*x^(1/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1 
/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/ 
2))/b^(7/4)/(b*x^3+a*x)^(1/2)-4/65*a^(13/4)*x^(1/2)*(a^(1/2)+b^(1/2)*x)*(( 
b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x^( 
1/2)/a^(1/4)),1/2*2^(1/2))/b^(7/4)/(b*x^3+a*x)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.28 \[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\frac {2 x \sqrt {x \left (a+b x^2\right )} \left (\left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}}-a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{13 b \sqrt {1+\frac {b x^2}{a}}} \] Input:

Integrate[x*(a*x + b*x^3)^(3/2),x]
 

Output:

(2*x*Sqrt[x*(a + b*x^2)]*((a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] - a^2*Hypergeo 
metric2F1[-3/2, 3/4, 7/4, -((b*x^2)/a)]))/(13*b*Sqrt[1 + (b*x^2)/a])
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1927, 1927, 1930, 1938, 266, 834, 27, 761, 1510}

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.

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

\(\Big \downarrow \) 1927

\(\displaystyle \frac {6}{13} a \int x^2 \sqrt {b x^3+a x}dx+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

\(\Big \downarrow \) 1927

\(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \int \frac {x^3}{\sqrt {b x^3+a x}}dx+\frac {2}{9} x^3 \sqrt {a x+b x^3}\right )+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

\(\Big \downarrow \) 1930

\(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 x \sqrt {a x+b x^3}}{5 b}-\frac {3 a \int \frac {x}{\sqrt {b x^3+a x}}dx}{5 b}\right )+\frac {2}{9} x^3 \sqrt {a x+b x^3}\right )+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

\(\Big \downarrow \) 1938

\(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 x \sqrt {a x+b x^3}}{5 b}-\frac {3 a \sqrt {x} \sqrt {a+b x^2} \int \frac {\sqrt {x}}{\sqrt {b x^2+a}}dx}{5 b \sqrt {a x+b x^3}}\right )+\frac {2}{9} x^3 \sqrt {a x+b x^3}\right )+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 x \sqrt {a x+b x^3}}{5 b}-\frac {6 a \sqrt {x} \sqrt {a+b x^2} \int \frac {x}{\sqrt {b x^2+a}}d\sqrt {x}}{5 b \sqrt {a x+b x^3}}\right )+\frac {2}{9} x^3 \sqrt {a x+b x^3}\right )+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 x \sqrt {a x+b x^3}}{5 b}-\frac {6 a \sqrt {x} \sqrt {a+b x^2} \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{5 b \sqrt {a x+b x^3}}\right )+\frac {2}{9} x^3 \sqrt {a x+b x^3}\right )+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 x \sqrt {a x+b x^3}}{5 b}-\frac {6 a \sqrt {x} \sqrt {a+b x^2} \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{5 b \sqrt {a x+b x^3}}\right )+\frac {2}{9} x^3 \sqrt {a x+b x^3}\right )+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 x \sqrt {a x+b x^3}}{5 b}-\frac {6 a \sqrt {x} \sqrt {a+b x^2} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{5 b \sqrt {a x+b x^3}}\right )+\frac {2}{9} x^3 \sqrt {a x+b x^3}\right )+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {6}{13} a \left (\frac {2}{9} a \left (\frac {2 x \sqrt {a x+b x^3}}{5 b}-\frac {6 a \sqrt {x} \sqrt {a+b x^2} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}}{\sqrt {b}}\right )}{5 b \sqrt {a x+b x^3}}\right )+\frac {2}{9} x^3 \sqrt {a x+b x^3}\right )+\frac {2}{13} x^2 \left (a x+b x^3\right )^{3/2}\)

Input:

Int[x*(a*x + b*x^3)^(3/2),x]
 

Output:

(2*x^2*(a*x + b*x^3)^(3/2))/13 + (6*a*((2*x^3*Sqrt[a*x + b*x^3])/9 + (2*a* 
((2*x*Sqrt[a*x + b*x^3])/(5*b) - (6*a*Sqrt[x]*Sqrt[a + b*x^2]*(-((-((Sqrt[ 
x]*Sqrt[a + b*x^2])/(Sqrt[a] + Sqrt[b]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x 
)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sq 
rt[x])/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^2]))/Sqrt[b]) + (a^(1/4)*(Sqr 
t[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*Ar 
cTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^2])))/(5*b* 
Sqrt[a*x + b*x^3])))/9))/13
 

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 1927

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

rule 1930

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p 
+ 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1)))   I 
nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, 
x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt 
Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {2 x^{2} \left (15 b^{2} x^{4}+25 a b \,x^{2}+4 a^{2}\right ) \left (b \,x^{2}+a \right )}{195 b \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{65 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(210\)
default	\(\frac {2 b \,x^{5} \sqrt {b \,x^{3}+a x}}{13}+\frac {10 a \,x^{3} \sqrt {b \,x^{3}+a x}}{39}+\frac {8 a^{2} x \sqrt {b \,x^{3}+a x}}{195 b}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{65 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(217\)
elliptic	\(\frac {2 b \,x^{5} \sqrt {b \,x^{3}+a x}}{13}+\frac {10 a \,x^{3} \sqrt {b \,x^{3}+a x}}{39}+\frac {8 a^{2} x \sqrt {b \,x^{3}+a x}}{195 b}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{65 b^{2} \sqrt {b \,x^{3}+a x}}\)	\(217\)

Input:

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

Output:

2/195*x^2*(15*b^2*x^4+25*a*b*x^2+4*a^2)/b*(b*x^2+a)/(x*(b*x^2+a))^(1/2)-4/ 
65*a^3/b^2*(-a*b)^(1/2)*((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-( 
-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)/(b*x^3+a* 
x)^(1/2)*(-2*(-a*b)^(1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^ 
(1/2),1/2*2^(1/2))+(-a*b)^(1/2)/b*EllipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/ 
2)*b)^(1/2),1/2*2^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.22 \[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\frac {2 \, {\left (12 \, a^{3} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (15 \, b^{3} x^{5} + 25 \, a b^{2} x^{3} + 4 \, a^{2} b x\right )} \sqrt {b x^{3} + a x}\right )}}{195 \, b^{2}} \] Input:

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

Output:

2/195*(12*a^3*sqrt(b)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/ 
b, 0, x)) + (15*b^3*x^5 + 25*a*b^2*x^3 + 4*a^2*b*x)*sqrt(b*x^3 + a*x))/b^2
 

Sympy [F]

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

integrate(x*(b*x**3+a*x)**(3/2),x)
 

Output:

Integral(x*(x*(a + b*x**2))**(3/2), x)
 

Maxima [F]

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

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

Output:

integrate((b*x^3 + a*x)^(3/2)*x, x)
 

Giac [F]

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

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

Output:

integrate((b*x^3 + a*x)^(3/2)*x, x)
 

Mupad [F(-1)]

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

int(x*(a*x + b*x^3)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int(x*(a*x + b*x^3)^(3/2), x)
 

Reduce [F]

\[ \int x \left (a x+b x^3\right )^{3/2} \, dx=\frac {\frac {8 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{2} x}{195}+\frac {10 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a b \,x^{3}}{39}+\frac {2 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b^{2} x^{5}}{13}-\frac {4 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{2}+a}d x \right ) a^{3}}{65}}{b} \] Input:

int(x*(b*x^3+a*x)^(3/2),x)
 

Output:

(2*(4*sqrt(x)*sqrt(a + b*x**2)*a**2*x + 25*sqrt(x)*sqrt(a + b*x**2)*a*b*x* 
*3 + 15*sqrt(x)*sqrt(a + b*x**2)*b**2*x**5 - 6*int((sqrt(x)*sqrt(a + b*x** 
2))/(a + b*x**2),x)*a**3))/(195*b)