Integrand size = 19, antiderivative size = 332 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx=-\frac {2 a \sqrt {a+b x^2}}{9 c (c x)^{9/2}}-\frac {22 b \sqrt {a+b x^2}}{45 c^3 (c x)^{5/2}}-\frac {8 b^2 \sqrt {a+b x^2}}{15 a c^5 \sqrt {c x}}+\frac {8 b^{5/2} \sqrt {c x} \sqrt {a+b x^2}}{15 a c^6 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {8 b^{9/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt {a+b x^2}}+\frac {4 b^{9/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{15 a^{3/4} c^{11/2} \sqrt {a+b x^2}} \] Output:
-2/9*a*(b*x^2+a)^(1/2)/c/(c*x)^(9/2)-22/45*b*(b*x^2+a)^(1/2)/c^3/(c*x)^(5/ 2)-8/15*b^2*(b*x^2+a)^(1/2)/a/c^5/(c*x)^(1/2)+8/15*b^(5/2)*(c*x)^(1/2)*(b* x^2+a)^(1/2)/a/c^6/(a^(1/2)+b^(1/2)*x)-8/15*b^(9/4)*(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)*(c*x) ^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1/2))/a^(3/4)/c^(11/2)/(b*x^2+a)^(1/2)+4/1 5*b^(9/4)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*Inve rseJacobiAM(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)),1/2*2^(1/2))/a^( 3/4)/c^(11/2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.17 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx=-\frac {2 a x \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{2},-\frac {5}{4},-\frac {b x^2}{a}\right )}{9 (c x)^{11/2} \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[(a + b*x^2)^(3/2)/(c*x)^(11/2),x]
Output:
(-2*a*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^2)/a)]) /(9*(c*x)^(11/2)*Sqrt[1 + (b*x^2)/a])
Time = 0.39 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {247, 247, 264, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {2 b \int \frac {\sqrt {b x^2+a}}{(c x)^{7/2}}dx}{3 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \int \frac {1}{(c x)^{3/2} \sqrt {b x^2+a}}dx}{5 c^2}-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}\right )}{3 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {b \int \frac {\sqrt {c x}}{\sqrt {b x^2+a}}dx}{a c^2}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 c^2}-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}\right )}{3 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 b \int \frac {c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 c^2}-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}\right )}{3 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 b \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\sqrt {a} c \int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {a} c \sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 c^2}-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}\right )}{3 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 b \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 c^2}-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}\right )}{3 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 c^2}-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}\right )}{3 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {c^2 \sqrt {c x} \sqrt {a+b x^2}}{\sqrt {a} c+\sqrt {b} c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 c^2}-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}\right )}{3 c^2}-\frac {2 \left (a+b x^2\right )^{3/2}}{9 c (c x)^{9/2}}\) |
Input:
Int[(a + b*x^2)^(3/2)/(c*x)^(11/2),x]
Output:
(-2*(a + b*x^2)^(3/2))/(9*c*(c*x)^(9/2)) + (2*b*((-2*Sqrt[a + b*x^2])/(5*c *(c*x)^(5/2)) + (2*b*((-2*Sqrt[a + b*x^2])/(a*c*Sqrt[c*x]) + (2*b*(-((-((c ^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(Sqrt[a]*c + Sqrt[b]*c*x)) + (a^(1/4)*Sqrt[c ]*(Sqrt[a]*c + Sqrt[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt[b]* c*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/( b^(1/4)*Sqrt[a + b*x^2]))/Sqrt[b]) + (a^(1/4)*Sqrt[c]*(Sqrt[a]*c + Sqrt[b] *c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt[b]*c*x)^2]*EllipticF[2*Ar cTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(2*b^(3/4)*Sqrt[a + b*x ^2])))/(a*c^3)))/(5*c^2)))/(3*c^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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]
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]
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]
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]
Time = 1.02 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(\frac {\frac {8 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}}{15}-\frac {4 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}}{15}-\frac {8 b^{3} x^{6}}{15}-\frac {46 a \,b^{2} x^{4}}{45}-\frac {32 a^{2} b \,x^{2}}{45}-\frac {2 a^{3}}{9}}{x^{4} \sqrt {b \,x^{2}+a}\, a \,c^{5} \sqrt {c x}}\) | \(234\) |
| risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (12 b^{2} x^{4}+11 a b \,x^{2}+5 a^{2}\right )}{45 x^{4} a \,c^{5} \sqrt {c x}}+\frac {4 b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{15 a \sqrt {b c \,x^{3}+a c x}\, c^{5} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(240\) |
| elliptic | \(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (-\frac {2 a \sqrt {b c \,x^{3}+a c x}}{9 c^{6} x^{5}}-\frac {22 b \sqrt {b c \,x^{3}+a c x}}{45 c^{6} x^{3}}-\frac {8 \left (x^{2} b c +a c \right ) b^{2}}{15 a \,c^{6} \sqrt {x \left (x^{2} b c +a c \right )}}+\frac {4 b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 a \,c^{5} \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(274\) |
Input:
int((b*x^2+a)^(3/2)/(c*x)^(11/2),x,method=_RETURNVERBOSE)
Output:
2/45/x^4*(12*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b) ^(1/2))/(-a*b)^(1/2))^(1/2)*(-b/(-a*b)^(1/2)*x)^(1/2)*EllipticE(((b*x+(-a* b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2*x^4-6*((b*x+(-a*b)^(1/2)) /(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-b/ (-a*b)^(1/2)*x)^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/ 2*2^(1/2))*a*b^2*x^4-12*b^3*x^6-23*a*b^2*x^4-16*a^2*b*x^2-5*a^3)/(b*x^2+a) ^(1/2)/a/c^5/(c*x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx=-\frac {2 \, {\left (12 \, \sqrt {b c} b^{2} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (12 \, b^{2} x^{4} + 11 \, a b x^{2} + 5 \, a^{2}\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{45 \, a c^{6} x^{5}} \] Input:
integrate((b*x^2+a)^(3/2)/(c*x)^(11/2),x, algorithm="fricas")
Output:
-2/45*(12*sqrt(b*c)*b^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^2 + a)*sqrt(c *x))/(a*c^6*x^5)
Result contains complex when optimal does not.
Time = 46.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.16 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx=\frac {a^{\frac {3}{2}} \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {3}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac {11}{2}} x^{\frac {9}{2}} \Gamma \left (- \frac {5}{4}\right )} \] Input:
integrate((b*x**2+a)**(3/2)/(c*x)**(11/2),x)
Output:
a**(3/2)*gamma(-9/4)*hyper((-9/4, -3/2), (-5/4,), b*x**2*exp_polar(I*pi)/a )/(2*c**(11/2)*x**(9/2)*gamma(-5/4))
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(c*x)^(11/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)/(c*x)^(11/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(c*x)^(11/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)/(c*x)^(11/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (c\,x\right )}^{11/2}} \,d x \] Input:
int((a + b*x^2)^(3/2)/(c*x)^(11/2),x)
Output:
int((a + b*x^2)^(3/2)/(c*x)^(11/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{(c x)^{11/2}} \, dx=\frac {2 \sqrt {c}\, \left (-\sqrt {b \,x^{2}+a}\, a -7 \sqrt {b \,x^{2}+a}\, b \,x^{2}+6 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{8}+a \,x^{6}}d x \right ) a^{2} x^{4}\right )}{21 \sqrt {x}\, c^{6} x^{4}} \] Input:
int((b*x^2+a)^(3/2)/(c*x)^(11/2),x)
Output:
(2*sqrt(c)*( - sqrt(a + b*x**2)*a - 7*sqrt(a + b*x**2)*b*x**2 + 6*sqrt(x)* int((sqrt(x)*sqrt(a + b*x**2))/(a*x**6 + b*x**8),x)*a**2*x**4))/(21*sqrt(x )*c**6*x**4)