\(\int \frac {(e x)^{3/2} (a+b x^2)^2}{(c+d x^2)^{3/2}} \, dx\) [851]

Optimal result

Integrand size = 28, antiderivative size = 245 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e \sqrt {e x} \sqrt {c+d x^2}}{21 c d^3}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}+\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt {c+d x^2}} \]

[Out]

(-a*d+b*c)^2*(e*x)^(5/2)/c/d^2/e/(d*x^2+c)^(1/2)+2/7*b^2*(e*x)^(5/2)*(d*x^2+c)^(1/2)/d^2/e-1/21*(21*a^2*d^2-70
*a*b*c*d+45*b^2*c^2)*e*(e*x)^(1/2)*(d*x^2+c)^(1/2)/c/d^3+1/42*(21*a^2*d^2-70*a*b*c*d+45*b^2*c^2)*e^(3/2)*(cos(
2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*Ell
ipticF(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)
+x*d^(1/2))^2)^(1/2)/c^(1/4)/d^(13/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {474, 470, 327, 335, 226} \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt {c+d x^2}}-\frac {e \sqrt {e x} \sqrt {c+d x^2} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right )}{21 c d^3}+\frac {(e x)^{5/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e} \]

[In]

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

[Out]

((b*c - a*d)^2*(e*x)^(5/2))/(c*d^2*e*Sqrt[c + d*x^2]) - ((45*b^2*c^2 - 70*a*b*c*d + 21*a^2*d^2)*e*Sqrt[e*x]*Sq
rt[c + d*x^2])/(21*c*d^3) + (2*b^2*(e*x)^(5/2)*Sqrt[c + d*x^2])/(7*d^2*e) + ((45*b^2*c^2 - 70*a*b*c*d + 21*a^2
*d^2)*e^(3/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt
[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(42*c^(1/4)*d^(13/4)*Sqrt[c + d*x^2])

Rule 226

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[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 470

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] - Dist[(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]

Rule 474

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(-(b*c - a*
d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b^2*e*n*(p + 1))), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a +
b*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a
, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {(e x)^{3/2} \left (\frac {1}{2} \left (-2 a^2 d^2+5 (b c-a d)^2\right )-b^2 c d x^2\right )}{\sqrt {c+d x^2}} \, dx}{c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx}{14 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e \sqrt {e x} \sqrt {c+d x^2}}{21 c d^3}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}+\frac {\left (\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{42 d^3} \\ & = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e \sqrt {e x} \sqrt {c+d x^2}}{21 c d^3}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}+\frac {\left (\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 d^3} \\ & = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e \sqrt {e x} \sqrt {c+d x^2}}{21 c d^3}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}+\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.78 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {e \sqrt {e x} \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} \left (-21 a^2 d^2+14 a b d \left (5 c+2 d x^2\right )-3 b^2 \left (15 c^2+6 c d x^2-2 d^2 x^4\right )\right )+i \left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )\right )}{21 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^3 \sqrt {c+d x^2}} \]

[In]

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

[Out]

(e*Sqrt[e*x]*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(-21*a^2*d^2 + 14*a*b*d*(5*c + 2*d*x^2) - 3*b^2*(15*c^2 + 6*c*d*x^2 -
2*d^2*x^4)) + I*(45*b^2*c^2 - 70*a*b*c*d + 21*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I
*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/(21*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.71 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.35

method	result	size
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (-\frac {e^{2} x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e \,x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d^{2}}+\frac {2 \left (\frac {b \left (2 a d -b c \right ) e^{2}}{d^{2}}-\frac {5 b^{2} e^{2} c}{7 d^{2}}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e^{2}}{2 d^{3}}-\frac {\left (\frac {b \left (2 a d -b c \right ) e^{2}}{d^{2}}-\frac {5 b^{2} e^{2} c}{7 d^{2}}\right ) c}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(331\)
default	\(\frac {e \sqrt {e x}\, \left (21 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2}-70 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d +45 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}+12 b^{2} d^{3} x^{5}+56 a b \,d^{3} x^{3}-36 b^{2} c \,d^{2} x^{3}-42 x \,a^{2} d^{3}+140 x a b c \,d^{2}-90 x \,b^{2} c^{2} d \right )}{42 x \sqrt {d \,x^{2}+c}\, d^{4}}\)	\(363\)
risch	\(\frac {2 b \left (3 b d \,x^{2}+14 a d -12 b c \right ) x \sqrt {d \,x^{2}+c}\, e^{2}}{21 d^{3} \sqrt {e x}}+\frac {\left (\frac {21 a^{2} d \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}+\frac {33 b^{2} c^{2} \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}-\frac {56 a b c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}-21 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{2} \sqrt {e x \left (d \,x^{2}+c \right )}}{21 d^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(572\)

[In]

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

[Out]

(e*x*(d*x^2+c))^(1/2)/e/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-1/d^3*e^2*x*(a^2*d^2-2*a*b*c*d+b^2*c^2)/((x^2+c/d)*d*e
*x)^(1/2)+2/7*b^2/d^2*e*x^2*(d*e*x^3+c*e*x)^(1/2)+2/3*(b/d^2*(2*a*d-b*c)*e^2-5/7*b^2/d^2*e^2*c)/d/e*(d*e*x^3+c
*e*x)^(1/2)+(1/2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*e^2/d^3-1/3*(b/d^2*(2*a*d-b*c)*e^2-5/7*b^2/d^2*e^2*c)/d*c)*(-c*d)
^(1/2)/d*((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)*(-x/(-c*d)^(1/
2)*d)^(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)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.71 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (45 \, b^{2} c^{2} d - 70 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} e x^{2} + {\left (45 \, b^{2} c^{3} - 70 \, a b c^{2} d + 21 \, a^{2} c d^{2}\right )} e\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (6 \, b^{2} d^{3} e x^{4} - 2 \, {\left (9 \, b^{2} c d^{2} - 14 \, a b d^{3}\right )} e x^{2} - {\left (45 \, b^{2} c^{2} d - 70 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} e\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{21 \, {\left (d^{5} x^{2} + c d^{4}\right )}} \]

[In]

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

[Out]

1/21*(((45*b^2*c^2*d - 70*a*b*c*d^2 + 21*a^2*d^3)*e*x^2 + (45*b^2*c^3 - 70*a*b*c^2*d + 21*a^2*c*d^2)*e)*sqrt(d
*e)*weierstrassPInverse(-4*c/d, 0, x) + (6*b^2*d^3*e*x^4 - 2*(9*b^2*c*d^2 - 14*a*b*d^3)*e*x^2 - (45*b^2*c^2*d
- 70*a*b*c*d^2 + 21*a^2*d^3)*e)*sqrt(d*x^2 + c)*sqrt(e*x))/(d^5*x^2 + c*d^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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