\(\int \frac {(e x)^{5/2} (a+b x^2)^2}{(c+d x^2)^{5/2}} \, dx\) [858]

Optimal result

Integrand size = 28, antiderivative size = 442 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{10 c d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt {c+d x^2}} \]

[Out]

1/3*(-a*d+b*c)^2*(e*x)^(7/2)/c/d^2/e/(d*x^2+c)^(3/2)+1/30*(5*a^2*d^2-70*a*b*c*d+77*b^2*c^2)*e*(e*x)^(3/2)/c/d^
3/(d*x^2+c)^(1/2)+2/5*b^2*(e*x)^(7/2)/d^2/e/(d*x^2+c)^(1/2)-1/10*(5*a^2*d^2-70*a*b*c*d+77*b^2*c^2)*e^2*(e*x)^(
1/2)*(d*x^2+c)^(1/2)/c/d^(7/2)/(c^(1/2)+x*d^(1/2))+1/10*(5*a^2*d^2-70*a*b*c*d+77*b^2*c^2)*e^(5/2)*(cos(2*arcta
n(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)))*EllipticE(
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^(3/4)/d^(15/4)/(d*x^2+c)^(1/2)-1/20*(5*a^2*d^2-70*a*b*c*d+77*b^2*c^2)*e^(5/2)*(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)))*EllipticF(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^(3/4)/d^(15/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {474, 470, 294, 335, 311, 226, 1210} \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt {c+d x^2}}+\frac {e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{10 c d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {e (e x)^{3/2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{30 c d^3 \sqrt {c+d x^2}}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}} \]

[In]

Int[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

((b*c - a*d)^2*(e*x)^(7/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e*(e*x)^(3/
2))/(30*c*d^3*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(7/2))/(5*d^2*e*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 70*a*b*c*d + 5
*a^2*d^2)*e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(10*c*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + ((77*b^2*c^2 - 70*a*b*c*d + 5*
a^2*d^2)*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*S
qrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(10*c^(3/4)*d^(15/4)*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*
d^2)*e^(5/2)*(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])/(20*c^(3/4)*d^(15/4)*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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 311

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

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]

Rule 1210

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
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {(e x)^{5/2} \left (\frac {1}{2} \left (-6 a^2 d^2+7 (b c-a d)^2\right )-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) \int \frac {(e x)^{5/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{30 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{20 c d^3} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{10 c d^3} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{10 \sqrt {c} d^{7/2}}+\frac {\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{10 \sqrt {c} d^{7/2}} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{10 c d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \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 )}{20 c^{3/4} d^{15/4} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 11.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.35 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {e (e x)^{3/2} \left (5 a^2 d^2 \left (c+3 d x^2\right )-10 a b c d \left (7 c+9 d x^2\right )+b^2 c \left (77 c^2+99 c d x^2+12 d^2 x^4\right )-3 \left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \left (c+d x^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{30 c d^3 \left (c+d x^2\right )^{3/2}} \]

[In]

Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

(e*(e*x)^(3/2)*(5*a^2*d^2*(c + 3*d*x^2) - 10*a*b*c*d*(7*c + 9*d*x^2) + b^2*c*(77*c^2 + 99*c*d*x^2 + 12*d^2*x^4
) - 3*(77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*(c + d*x^2)*Hypergeometric2F1[-1/4, 1/2, 3/4,
-(c/(d*x^2))]))/(30*c*d^3*(c + d*x^2)^(3/2))

Maple [A] (verified)

Time = 4.73 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.88

method	result	size
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (-\frac {e^{2} x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d^{5} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {e^{3} x^{2} \left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right )}{2 d^{3} c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e^{2} x \sqrt {d e \,x^{3}+c e x}}{5 d^{3}}+\frac {\left (\frac {2 \left (a d -b c \right ) b \,e^{3}}{d^{3}}-\frac {e^{3} \left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right )}{4 d^{3} c}-\frac {3 b^{2} e^{3} c}{5 d^{3}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(387\)
risch	\(\frac {2 b^{2} x^{2} \sqrt {d \,x^{2}+c}\, e^{3}}{5 d^{3} \sqrt {e x}}+\frac {\left (\frac {b \left (10 a d -13 b c \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}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\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}\right )}{d \sqrt {d e \,x^{3}+c e x}}+\left (5 a^{2} d^{2}-20 a b c d +15 b^{2} c^{2}\right ) \left (\frac {x^{2}}{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}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\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}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )-5 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x \sqrt {d e \,x^{3}+c e x}}{3 c e \,d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {x^{2}}{2 c^{2} \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}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\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}\right )}{4 c^{2} d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{3} \sqrt {e x \left (d \,x^{2}+c \right )}}{5 d^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(691\)
default	\(\text {Expression too large to display}\)	\(1191\)

[In]

int((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

(e*x*(d*x^2+c))^(1/2)/e/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-1/3*e^2/d^5*x*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*e*x^3+c*e
*x)^(1/2)/(x^2+c/d)^2+1/2/d^3*e^3*x^2/c*(a^2*d^2-6*a*b*c*d+5*b^2*c^2)/((x^2+c/d)*d*e*x)^(1/2)+2/5*b^2/d^3*e^2*
x*(d*e*x^3+c*e*x)^(1/2)+(2*(a*d-b*c)*b*e^3/d^3-1/4/d^3/c*e^3*(a^2*d^2-6*a*b*c*d+5*b^2*c^2)-3/5*b^2/d^3*e^3*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)*(-2*(-c*d)^(1/2)/d*EllipticE(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)
,1/2*2^(1/2))+(-c*d)^(1/2)/d*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 = 0.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.59 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {3 \, {\left ({\left (77 \, b^{2} c^{2} d^{2} - 70 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} e^{2} x^{4} + 2 \, {\left (77 \, b^{2} c^{3} d - 70 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} e^{2} x^{2} + {\left (77 \, b^{2} c^{4} - 70 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} e^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (12 \, b^{2} c d^{3} e^{2} x^{5} + 3 \, {\left (33 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} e^{2} x^{3} + {\left (77 \, b^{2} c^{3} d - 70 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} e^{2} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{30 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}} \]

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

1/30*(3*((77*b^2*c^2*d^2 - 70*a*b*c*d^3 + 5*a^2*d^4)*e^2*x^4 + 2*(77*b^2*c^3*d - 70*a*b*c^2*d^2 + 5*a^2*c*d^3)
*e^2*x^2 + (77*b^2*c^4 - 70*a*b*c^3*d + 5*a^2*c^2*d^2)*e^2)*sqrt(d*e)*weierstrassZeta(-4*c/d, 0, weierstrassPI
nverse(-4*c/d, 0, x)) + (12*b^2*c*d^3*e^2*x^5 + 3*(33*b^2*c^2*d^2 - 30*a*b*c*d^3 + 5*a^2*d^4)*e^2*x^3 + (77*b^
2*c^3*d - 70*a*b*c^2*d^2 + 5*a^2*c*d^3)*e^2*x)*sqrt(d*x^2 + c)*sqrt(e*x))/(c*d^6*x^4 + 2*c^2*d^5*x^2 + c^3*d^4
)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x)**(5/2)*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(5/2), x)

Giac [F]

\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x)

[Out]

int(((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x)