3.856 \(\int \frac{(a+b x^2)^2}{(e x)^{7/2} (c+d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=434 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{(e x)^{3/2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}} \]

[Out]

(-2*a^2)/(5*c*e*(e*x)^(5/2)*Sqrt[c + d*x^2]) - (2*a*(10*b*c - 7*a*d))/(5*c^2*e^3*Sqrt[e*x]*Sqrt[c + d*x^2]) +
((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(e*x)^(3/2))/(5*c^3*e^5*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 3*a*d*(10*b*c -
7*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(5*c^3*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) + ((5*b^2*c^2 - 3*a*d*(10*b*c -
7*a*d))*(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])/(5*c^(11/4)*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d)
)*(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])/(10*c^(11/4)*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.422497, antiderivative size = 434, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {462, 453, 290, 329, 305, 220, 1196} \[ -\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{(e x)^{3/2} \left (5 b^2 c^2-3 a d (10 b c-7 a d)\right )}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]

[Out]

(-2*a^2)/(5*c*e*(e*x)^(5/2)*Sqrt[c + d*x^2]) - (2*a*(10*b*c - 7*a*d))/(5*c^2*e^3*Sqrt[e*x]*Sqrt[c + d*x^2]) +
((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*(e*x)^(3/2))/(5*c^3*e^5*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 3*a*d*(10*b*c -
7*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(5*c^3*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) + ((5*b^2*c^2 - 3*a*d*(10*b*c -
7*a*d))*(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])/(5*c^(11/4)*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d)
)*(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])/(10*c^(11/4)*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2])

Rule 462

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

Rule 453

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

Rule 290

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

Rule 329

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 305

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 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

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)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx &=-\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}+\frac{2 \int \frac{\frac{1}{2} a (10 b c-7 a d)+\frac{5}{2} b^2 c x^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx}{5 c e^2}\\ &=-\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}}+\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \int \frac{\sqrt{e x}}{\left (c+d x^2\right )^{3/2}} \, dx}{5 c^2 e^4}\\ &=-\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}}+\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) (e x)^{3/2}}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{10 c^3 e^4}\\ &=-\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}}+\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) (e x)^{3/2}}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 c^3 e^5}\\ &=-\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}}+\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) (e x)^{3/2}}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 c^{5/2} \sqrt{d} e^4}+\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \operatorname{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 )}{5 c^{5/2} \sqrt{d} e^4}\\ &=-\frac{2 a^2}{5 c e (e x)^{5/2} \sqrt{c+d x^2}}-\frac{2 a (10 b c-7 a d)}{5 c^2 e^3 \sqrt{e x} \sqrt{c+d x^2}}+\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) (e x)^{3/2}}{5 c^3 e^5 \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \sqrt{e x} \sqrt{c+d x^2}}{5 c^3 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \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 )}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \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 )}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.12478, size = 141, normalized size = 0.32 \[ \frac{x \left (x^4 \sqrt{\frac{d x^2}{c}+1} \left (-21 a^2 d^2+30 a b c d-5 b^2 c^2\right ) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{d x^2}{c}\right )+a^2 \left (-6 c^2+42 c d x^2+63 d^2 x^4\right )-30 a b c x^2 \left (2 c+3 d x^2\right )+15 b^2 c^2 x^4\right )}{15 c^3 (e x)^{7/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]

[Out]

(x*(15*b^2*c^2*x^4 - 30*a*b*c*x^2*(2*c + 3*d*x^2) + a^2*(-6*c^2 + 42*c*d*x^2 + 63*d^2*x^4) + (-5*b^2*c^2 + 30*
a*b*c*d - 21*a^2*d^2)*x^4*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)]))/(15*c^3*(e*x)^(
7/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.025, size = 638, normalized size = 1.5 \begin{align*} -{\frac{1}{10\,d{x}^{2}{e}^{3}{c}^{3}} \left ( 42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}-60\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d+10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}+30\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d-5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-42\,{x}^{4}{a}^{2}{d}^{3}+60\,{x}^{4}abc{d}^{2}-10\,{x}^{4}{b}^{2}{c}^{2}d-28\,{x}^{2}{a}^{2}c{d}^{2}+40\,{x}^{2}ab{c}^{2}d+4\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x)

[Out]

-1/10/x^2*(42*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c
*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*c*d^2-60*((d*x+(-c*d
)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*Ellipt
icE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b*c^2*d+10*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1
/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(
-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^2*c^3-21*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(
1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1
/2))*x^2*a^2*c*d^2+30*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)
*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b*c^2*d-5*((d*
x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)
*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^2*c^3-42*x^4*a^2*d^3+60*x^4*a*b*c*d^2-10
*x^4*b^2*c^2*d-28*x^2*a^2*c*d^2+40*x^2*a*b*c^2*d+4*a^2*c^2*d)/(d*x^2+c)^(1/2)/d/e^3/(e*x)^(1/2)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}{d^{2} e^{4} x^{8} + 2 \, c d e^{4} x^{6} + c^{2} e^{4} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)*sqrt(e*x)/(d^2*e^4*x^8 + 2*c*d*e^4*x^6 + c^2*e^4*x^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/(e*x)**(7/2)/(d*x**2+c)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)), x)