Integrand size = 28, antiderivative size = 234 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx=-\frac {2 \left (b^2 c^2-7 a d (2 b c+a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{21 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e^3}-\frac {2 \left (b^2 c^2-7 a d (2 b c+a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{5/4} e^{5/2} \sqrt {c+d x^2}} \] Output:
-2/21*(b^2*c^2-7*a*d*(a*d+2*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1/2)/c/d/e^3-2/3* a^2*(d*x^2+c)^(3/2)/c/e/(e*x)^(3/2)+2/7*b^2*(e*x)^(1/2)*(d*x^2+c)^(3/2)/d/ e^3-2/21*(b^2*c^2-7*a*d*(a*d+2*b*c))*(c^(1/2)+d^(1/2)*x)*((d*x^2+c)/(c^(1/ 2)+d^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4 )/e^(1/2)),1/2*2^(1/2))/c^(1/4)/d^(5/4)/e^(5/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.15 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx=\frac {x^{5/2} \left (\frac {2 \left (c+d x^2\right ) \left (-7 a^2 d+14 a b d x^2+b^2 x^2 \left (2 c+3 d x^2\right )\right )}{d x^{3/2}}+\frac {4 i \left (-b^2 c^2+14 a b c d+7 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d}\right )}{21 (e x)^{5/2} \sqrt {c+d x^2}} \] Input:
Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(5/2),x]
Output:
(x^(5/2)*((2*(c + d*x^2)*(-7*a^2*d + 14*a*b*d*x^2 + b^2*x^2*(2*c + 3*d*x^2 )))/(d*x^(3/2)) + ((4*I)*(-(b^2*c^2) + 14*a*b*c*d + 7*a^2*d^2)*Sqrt[1 + c/ (d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(S qrt[(I*Sqrt[c])/Sqrt[d]]*d)))/(21*(e*x)^(5/2)*Sqrt[c + d*x^2])
Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {365, 27, 363, 248, 266, 761}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {3 \left (b^2 c x^2+a (2 b c+a d)\right ) \sqrt {d x^2+c}}{2 \sqrt {e x}}dx}{3 c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b^2 c x^2+a (2 b c+a d)\right ) \sqrt {d x^2+c}}{\sqrt {e x}}dx}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {2 b^2 c \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e}-\frac {\left (b^2 c^2-7 a d (a d+2 b c)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {e x}}dx}{7 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {2 b^2 c \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e}-\frac {\left (b^2 c^2-7 a d (a d+2 b c)\right ) \left (\frac {2}{3} c \int \frac {1}{\sqrt {e x} \sqrt {d x^2+c}}dx+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )}{7 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {2 b^2 c \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e}-\frac {\left (b^2 c^2-7 a d (a d+2 b c)\right ) \left (\frac {4 c \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{3 e}+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )}{7 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {2 b^2 c \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e}-\frac {\left (b^2 c^2-7 a d (a d+2 b c)\right ) \left (\frac {2 c^{3/4} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{d} e^{3/2} \sqrt {c+d x^2}}+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )}{7 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}\) |
Input:
Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(5/2),x]
Output:
(-2*a^2*(c + d*x^2)^(3/2))/(3*c*e*(e*x)^(3/2)) + ((2*b^2*c*Sqrt[e*x]*(c + d*x^2)^(3/2))/(7*d*e) - ((b^2*c^2 - 7*a*d*(2*b*c + a*d))*((2*Sqrt[e*x]*Sqr t[c + d*x^2])/(3*e) + (2*c^(3/4)*(Sqrt[c]*e + Sqrt[d]*e*x)*Sqrt[(c*e^2 + d *e^2*x^2)/(Sqrt[c]*e + Sqrt[d]*e*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e* x])/(c^(1/4)*Sqrt[e])], 1/2])/(3*d^(1/4)*e^(3/2)*Sqrt[c + d*x^2])))/(7*d)) /(c*e^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 + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && 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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
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]
Time = 0.82 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(-\frac {2 \sqrt {x^{2} d +c}\, \left (-3 d \,b^{2} x^{4}-14 a b d \,x^{2}-2 b^{2} c \,x^{2}+7 d \,a^{2}\right )}{21 d x \,e^{2} \sqrt {e x}}+\frac {2 \left (7 a^{2} d^{2}+14 a b c d -b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (x^{2} d +c \right )}}{21 d^{2} \sqrt {d e \,x^{3}+c e x}\, e^{2} \sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(222\) |
| elliptic | \(\frac {\sqrt {e x \left (x^{2} d +c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{3 e^{3} x^{2}}+\frac {2 b^{2} x^{2} \sqrt {d e \,x^{3}+c e x}}{7 e^{3}}+\frac {2 \left (\frac {b \left (2 a d +b c \right )}{e^{2}}-\frac {5 b^{2} c}{7 e^{2}}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (\frac {a \left (a d +2 b c \right )}{e^{2}}-\frac {d \,a^{2}}{3 e^{2}}-\frac {\left (\frac {b \left (2 a d +b c \right )}{e^{2}}-\frac {5 b^{2} c}{7 e^{2}}\right ) c}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(282\) |
| default | \(\frac {\frac {2 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a^{2} d^{2} x}{3}+\frac {4 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a b c d x}{3}-\frac {2 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, b^{2} c^{2} x}{21}+\frac {2 b^{2} d^{3} x^{6}}{7}+\frac {4 a b \,d^{3} x^{4}}{3}+\frac {10 b^{2} c \,d^{2} x^{4}}{21}-\frac {2 a^{2} d^{3} x^{2}}{3}+\frac {4 a b c \,d^{2} x^{2}}{3}+\frac {4 b^{2} c^{2} d \,x^{2}}{21}-\frac {2 c \,a^{2} d^{2}}{3}}{\sqrt {x^{2} d +c}\, x \,e^{2} \sqrt {e x}\, d^{2}}\) | \(383\) |
Input:
int((b*x^2+a)^2*(d*x^2+c)^(1/2)/(e*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/21*(d*x^2+c)^(1/2)*(-3*b^2*d*x^4-14*a*b*d*x^2-2*b^2*c*x^2+7*a^2*d)/d/x/ e^2/(e*x)^(1/2)+2/21*(7*a^2*d^2+14*a*b*c*d-b^2*c^2)/d^2*(-c*d)^(1/2)*((x+( -c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(-2*(x-(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d )^(1/2)*(-d/(-c*d)^(1/2)*x)^(1/2)/(d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+(-c* d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2))/e^2*(e*x*(d*x^2+c))^(1/2)/( e*x)^(1/2)/(d*x^2+c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.46 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (b^{2} c^{2} - 14 \, a b c d - 7 \, a^{2} d^{2}\right )} \sqrt {d e} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (3 \, b^{2} d^{2} x^{4} - 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} c d + 7 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{21 \, d^{2} e^{3} x^{2}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/(e*x)^(5/2),x, algorithm="fricas")
Output:
-2/21*(2*(b^2*c^2 - 14*a*b*c*d - 7*a^2*d^2)*sqrt(d*e)*x^2*weierstrassPInve rse(-4*c/d, 0, x) - (3*b^2*d^2*x^4 - 7*a^2*d^2 + 2*(b^2*c*d + 7*a*b*d^2)*x ^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(d^2*e^3*x^2)
Result contains complex when optimal does not.
Time = 6.05 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx=\frac {a^{2} \sqrt {c} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {a b \sqrt {c} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {b^{2} \sqrt {c} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/(e*x)**(5/2),x)
Output:
a**2*sqrt(c)*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), d*x**2*exp_polar(I*pi )/c)/(2*e**(5/2)*x**(3/2)*gamma(1/4)) + a*b*sqrt(c)*sqrt(x)*gamma(1/4)*hyp er((-1/2, 1/4), (5/4,), d*x**2*exp_polar(I*pi)/c)/(e**(5/2)*gamma(5/4)) + b**2*sqrt(c)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), d*x**2*exp_pol ar(I*pi)/c)/(2*e**(5/2)*gamma(9/4))
\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/(e*x)^(5/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/(e*x)^(5/2), x)
\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/(e*x)^(5/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/(e*x)^(5/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{{\left (e\,x\right )}^{5/2}} \,d x \] Input:
int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/(e*x)^(5/2),x)
Output:
int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/(e*x)^(5/2), x)
\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx=\frac {2 \sqrt {e}\, \left (-21 \sqrt {d \,x^{2}+c}\, a^{2} d^{2}-28 \sqrt {d \,x^{2}+c}\, a b c d +14 \sqrt {d \,x^{2}+c}\, a b \,d^{2} x^{2}+2 \sqrt {d \,x^{2}+c}\, b^{2} c^{2}+2 \sqrt {d \,x^{2}+c}\, b^{2} c d \,x^{2}+3 \sqrt {d \,x^{2}+c}\, b^{2} d^{2} x^{4}-21 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{5}+c \,x^{3}}d x \right ) a^{2} c \,d^{2} x -42 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{5}+c \,x^{3}}d x \right ) a b \,c^{2} d x +3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{5}+c \,x^{3}}d x \right ) b^{2} c^{3} x \right )}{21 \sqrt {x}\, d^{2} e^{3} x} \] Input:
int((b*x^2+a)^2*(d*x^2+c)^(1/2)/(e*x)^(5/2),x)
Output:
(2*sqrt(e)*( - 21*sqrt(c + d*x**2)*a**2*d**2 - 28*sqrt(c + d*x**2)*a*b*c*d + 14*sqrt(c + d*x**2)*a*b*d**2*x**2 + 2*sqrt(c + d*x**2)*b**2*c**2 + 2*sq rt(c + d*x**2)*b**2*c*d*x**2 + 3*sqrt(c + d*x**2)*b**2*d**2*x**4 - 21*sqrt (x)*int((sqrt(x)*sqrt(c + d*x**2))/(c*x**3 + d*x**5),x)*a**2*c*d**2*x - 42 *sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c*x**3 + d*x**5),x)*a*b*c**2*d*x + 3*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c*x**3 + d*x**5),x)*b**2*c**3* x))/(21*sqrt(x)*d**2*e**3*x)