\(\int \frac {(a x+b x^3)^{3/2}}{x^7} \, dx\) [62]

Optimal result

Integrand size = 17, antiderivative size = 306 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^7} \, dx=\frac {8 b^{5/2} x \left (a+b x^2\right )}{15 a \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {4 b \sqrt {a x+b x^3}}{15 x^3}-\frac {8 b^2 \sqrt {a x+b x^3}}{15 a x}-\frac {2 \left (a x+b x^3\right )^{3/2}}{9 x^6}-\frac {8 b^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a x+b x^3}}+\frac {4 b^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a x+b x^3}} \] Output:

8/15*b^(5/2)*x*(b*x^2+a)/a/(a^(1/2)+b^(1/2)*x)/(b*x^3+a*x)^(1/2)-4/15*b*(b 
*x^3+a*x)^(1/2)/x^3-8/15*b^2*(b*x^3+a*x)^(1/2)/a/x-2/9*(b*x^3+a*x)^(3/2)/x 
^6-8/15*b^(9/4)*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))/a^ 
(3/4)/(b*x^3+a*x)^(1/2)+4/15*b^(9/4)*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))/a^(3/4)/(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 = 10.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.18 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^7} \, dx=-\frac {2 a \sqrt {x \left (a+b x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{2},-\frac {5}{4},-\frac {b x^2}{a}\right )}{9 x^5 \sqrt {1+\frac {b x^2}{a}}} \] Input:

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

Output:

(-2*a*Sqrt[x*(a + b*x^2)]*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^2)/a) 
])/(9*x^5*Sqrt[1 + (b*x^2)/a])
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {1926, 1926, 1931, 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.

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

\(\Big \downarrow \) 1926

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

\(\Big \downarrow \) 1926

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

\(\Big \downarrow \) 1931

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

\(\Big \downarrow \) 1938

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

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

Output:

(-2*(a*x + b*x^3)^(3/2))/(9*x^6) + (2*b*((-2*Sqrt[a*x + b*x^3])/(5*x^3) + 
(2*b*((-2*Sqrt[a*x + b*x^3])/(a*x) + (2*b*Sqrt[x]*Sqrt[a + b*x^2]*(-((-((S 
qrt[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 
)*Sqrt[x])/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^2]))/Sqrt[b]) + (a^(1/4)* 
(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[ 
2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^2])))/( 
a*Sqrt[a*x + b*x^3])))/5))/3
 

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 1926

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 + j*p + 1))), x] - Simp[b*p 
*((n - j)/(c^n*(m + j*p + 1)))   Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), 
 x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (Integer 
sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
 

rule 1931

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p 
+ 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*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]) && LtQ[ 
m + j*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 = 0.61 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {2 \left (b \,x^{2}+a \right ) \left (12 b^{2} x^{4}+11 a b \,x^{2}+5 a^{2}\right )}{45 x^{4} \sqrt {x \left (b \,x^{2}+a \right )}\, a}+\frac {4 b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 a \sqrt {b \,x^{3}+a x}}\)	\(210\)
default	\(-\frac {2 a \sqrt {b \,x^{3}+a x}}{9 x^{5}}-\frac {22 b \sqrt {b \,x^{3}+a x}}{45 x^{3}}-\frac {8 \left (b \,x^{2}+a \right ) b^{2}}{15 a \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {4 b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 a \sqrt {b \,x^{3}+a x}}\)	\(223\)
elliptic	\(-\frac {2 a \sqrt {b \,x^{3}+a x}}{9 x^{5}}-\frac {22 b \sqrt {b \,x^{3}+a x}}{45 x^{3}}-\frac {8 \left (b \,x^{2}+a \right ) b^{2}}{15 a \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {4 b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 a \sqrt {b \,x^{3}+a x}}\)	\(223\)

Input:

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

Output:

-2/45*(b*x^2+a)*(12*b^2*x^4+11*a*b*x^2+5*a^2)/x^4/(x*(b*x^2+a))^(1/2)/a+4/ 
15/a*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 = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^7} \, dx=-\frac {2 \, {\left (12 \, b^{\frac {5}{2}} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (12 \, b^{2} x^{4} + 11 \, a b x^{2} + 5 \, a^{2}\right )} \sqrt {b x^{3} + a x}\right )}}{45 \, a x^{5}} \] Input:

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

Output:

-2/45*(12*b^(5/2)*x^5*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/ 
b, 0, x)) + (12*b^2*x^4 + 11*a*b*x^2 + 5*a^2)*sqrt(b*x^3 + a*x))/(a*x^5)
 

Sympy [F]

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

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

Output:

Integral((x*(a + b*x**2))**(3/2)/x**7, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^7} \, dx=\frac {-\frac {2 \sqrt {b \,x^{2}+a}\, a}{21}-\frac {2 \sqrt {b \,x^{2}+a}\, b \,x^{2}}{3}+\frac {4 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{8}+a \,x^{6}}d x \right ) a^{2} x^{4}}{7}}{\sqrt {x}\, x^{4}} \] Input:

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

Output:

(2*( - sqrt(a + b*x**2)*a - 7*sqrt(a + b*x**2)*b*x**2 + 6*sqrt(x)*int((sqr 
t(x)*sqrt(a + b*x**2))/(a*x**6 + b*x**8),x)*a**2*x**4))/(21*sqrt(x)*x**4)