\(\int \frac {(e x)^{5/2} (a+b x^2)^2}{(c+d x^2)^{3/2}} \, dx\) [850]

Optimal result

Integrand size = 28, antiderivative size = 436 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{15 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\sqrt [4]{c} \left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{15/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{30 d^{15/4} \sqrt {c+d x^2}} \]

[Out]

(-a*d+b*c)^2*(e*x)^(7/2)/c/d^2/e/(d*x^2+c)^(1/2)-1/45*(45*a^2*d^2-126*a*b*c*d+77*b^2*c^2)*e*(e*x)^(3/2)*(d*x^2
+c)^(1/2)/c/d^3+2/9*b^2*(e*x)^(7/2)*(d*x^2+c)^(1/2)/d^2/e+1/15*(45*a^2*d^2-126*a*b*c*d+77*b^2*c^2)*e^2*(e*x)^(
1/2)*(d*x^2+c)^(1/2)/d^(7/2)/(c^(1/2)+x*d^(1/2))-1/15*c^(1/4)*(45*a^2*d^2-126*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)))*El
lipticE(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)/d^(15/4)/(d*x^2+c)^(1/2)+1/30*c^(1/4)*(45*a^2*d^2-126*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)))*Elli
pticF(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)/d^(15/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 436, 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, 327, 335, 311, 226, 1210} \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt [4]{c} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{30 d^{15/4} \sqrt {c+d x^2}}-\frac {\sqrt [4]{c} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{15/4} \sqrt {c+d x^2}}+\frac {e^2 \sqrt {e x} \sqrt {c+d x^2} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right )}{15 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {e (e x)^{3/2} \sqrt {c+d x^2} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right )}{45 c d^3}+\frac {(e x)^{7/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e} \]

[In]

Int[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

((b*c - a*d)^2*(e*x)^(7/2))/(c*d^2*e*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*e*(e*x)^(3/2)
*Sqrt[c + d*x^2])/(45*c*d^3) + (2*b^2*(e*x)^(7/2)*Sqrt[c + d*x^2])/(9*d^2*e) + ((77*b^2*c^2 - 126*a*b*c*d + 45
*a^2*d^2)*e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) - (c^(1/4)*(77*b^2*c^2 - 126*a*b*c
*d + 45*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)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(15*d^(15/4)*Sqrt[c + d*x^2]) + (c^(1/4)*(77*b^2*c^2 - 126*a*b*c*d
+ 45*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])/(30*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 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 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]

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}}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {(e x)^{5/2} \left (\frac {1}{2} \left (-2 a^2 d^2+7 (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)^{7/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) \int \frac {(e x)^{5/2}}{\sqrt {c+d x^2}} \, dx}{18 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{30 d^3} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (\left (77 b^2 c^2-126 a b c d+45 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 )}{15 d^3} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (\sqrt {c} \left (77 b^2 c^2-126 a b c d+45 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 )}{15 d^{7/2}}-\frac {\left (\sqrt {c} \left (77 b^2 c^2-126 a b c d+45 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 )}{15 d^{7/2}} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{15 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\sqrt [4]{c} \left (77 b^2 c^2-126 a b c d+45 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 )}{15 d^{15/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (77 b^2 c^2-126 a b c d+45 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 )}{30 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 = 11.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.31 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {e (e x)^{3/2} \left (-45 a^2 d^2+18 a b d \left (7 c+2 d x^2\right )+b^2 \left (-77 c^2-22 c d x^2+10 d^2 x^4\right )+3 \left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{45 d^3 \sqrt {c+d x^2}} \]

[In]

Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

(e*(e*x)^(3/2)*(-45*a^2*d^2 + 18*a*b*d*(7*c + 2*d*x^2) + b^2*(-77*c^2 - 22*c*d*x^2 + 10*d^2*x^4) + 3*(77*b^2*c
^2 - 126*a*b*c*d + 45*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(45*d^3*S
qrt[c + d*x^2])

Maple [A] (verified)

Time = 3.99 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.89

method	result	size
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (-\frac {e^{3} x^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e^{2} x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d^{2}}+\frac {2 \left (\frac {b \left (2 a d -b c \right ) e^{3}}{d^{2}}-\frac {7 b^{2} e^{3} c}{9 d^{2}}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e^{3}}{2 d^{3}}-\frac {3 \left (\frac {b \left (2 a d -b c \right ) e^{3}}{d^{2}}-\frac {7 b^{2} e^{3} c}{9 d^{2}}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(386\)
risch	\(\frac {2 b \,x^{2} \left (5 b d \,x^{2}+18 a d -16 b c \right ) \sqrt {d \,x^{2}+c}\, e^{3}}{45 d^{3} \sqrt {e x}}+\frac {\left (\frac {\left (15 a^{2} d^{2}-48 a b c d +31 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}-15 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x^{2}}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{3} \sqrt {e x \left (d \,x^{2}+c \right )}}{15 d^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(471\)
default	\(\frac {e^{2} \sqrt {e x}\, \left (20 b^{2} d^{3} x^{6}+270 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-756 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +462 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-135 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+378 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d -231 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}+72 a b \,d^{3} x^{4}-44 b^{2} c \,d^{2} x^{4}-90 a^{2} d^{3} x^{2}+252 a b c \,d^{2} x^{2}-154 b^{2} c^{2} d \,x^{2}\right )}{90 x \sqrt {d \,x^{2}+c}\, d^{4}}\)	\(618\)

[In]

int((e*x)^(5/2)*(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^3*x^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/((x^2+c/d)*d
*e*x)^(1/2)+2/9*b^2/d^2*e^2*x^3*(d*e*x^3+c*e*x)^(1/2)+2/5*(b/d^2*(2*a*d-b*c)*e^3-7/9*b^2/d^2*e^3*c)/d/e*x*(d*e
*x^3+c*e*x)^(1/2)+(3/2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*e^3/d^3-3/5*(b/d^2*(2*a*d-b*c)*e^3-7/9*b^2/d^2*e^3*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)*(-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 = 0.10 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.44 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {3 \, {\left ({\left (77 \, b^{2} c^{2} d - 126 \, a b c d^{2} + 45 \, a^{2} d^{3}\right )} e^{2} x^{2} + {\left (77 \, b^{2} c^{3} - 126 \, a b c^{2} d + 45 \, a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (10 \, b^{2} d^{3} e^{2} x^{5} - 2 \, {\left (11 \, b^{2} c d^{2} - 18 \, a b d^{3}\right )} e^{2} x^{3} - {\left (77 \, b^{2} c^{2} d - 126 \, a b c d^{2} + 45 \, a^{2} d^{3}\right )} e^{2} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{45 \, {\left (d^{5} x^{2} + c d^{4}\right )}} \]

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

-1/45*(3*((77*b^2*c^2*d - 126*a*b*c*d^2 + 45*a^2*d^3)*e^2*x^2 + (77*b^2*c^3 - 126*a*b*c^2*d + 45*a^2*c*d^2)*e^
2)*sqrt(d*e)*weierstrassZeta(-4*c/d, 0, weierstrassPInverse(-4*c/d, 0, x)) - (10*b^2*d^3*e^2*x^5 - 2*(11*b^2*c
*d^2 - 18*a*b*d^3)*e^2*x^3 - (77*b^2*c^2*d - 126*a*b*c*d^2 + 45*a^2*d^3)*e^2*x)*sqrt(d*x^2 + c)*sqrt(e*x))/(d^
5*x^2 + c*d^4)

Sympy [F]

\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (e x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((e*x)**(5/2)*(b*x**2+a)**2/(d*x**2+c)**(3/2),x)

[Out]

Integral((e*x)**(5/2)*(a + b*x**2)**2/(c + d*x**2)**(3/2), x)

Maxima [F]

\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(3/2), x)

Giac [F]

\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int(((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x)

[Out]

int(((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2), x)