3.834 \(\int \sqrt{e x} (a+b x^2)^2 (c+d x^2)^{3/2} \, dx\)

Optimal. Leaf size=482 \[ \frac{4 c^{9/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{3315 d^{11/4} \sqrt{c+d x^2}}+\frac{8 c^2 \sqrt{e x} \sqrt{c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{8 c^{9/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{11/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{1989 d^2 e}+\frac{4 c (e x)^{3/2} \sqrt{c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^2 e}-\frac{2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3} \]

[Out]

(4*c*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(3315*d^2*e) + (8*c^2*(221*a^2*d^2 +
3*b*c*(7*b*c - 34*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(3315*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(221*a^2*d^2 + 3*
b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(1989*d^2*e) - (2*b*(7*b*c - 34*a*d)*(e*x)^(3/2)*(c + d*x
^2)^(5/2))/(221*d^2*e) + (2*b^2*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(17*d*e^3) - (8*c^(9/4)*(221*a^2*d^2 + 3*b*c*(7
*b*c - 34*a*d))*Sqrt[e]*(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])/(3315*d^(11/4)*Sqrt[c + d*x^2]) + (4*c^(9/4)*(221*a^2*d^2 + 3*b*c*(
7*b*c - 34*a*d))*Sqrt[e]*(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])/(3315*d^(11/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.468646, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {464, 459, 279, 329, 305, 220, 1196} \[ \frac{8 c^2 \sqrt{e x} \sqrt{c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{4 c^{9/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{11/4} \sqrt{c+d x^2}}-\frac{8 c^{9/4} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{11/4} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{1989 d^2 e}+\frac{4 c (e x)^{3/2} \sqrt{c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^2 e}-\frac{2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*c*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(3315*d^2*e) + (8*c^2*(221*a^2*d^2 +
3*b*c*(7*b*c - 34*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(3315*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(221*a^2*d^2 + 3*
b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(1989*d^2*e) - (2*b*(7*b*c - 34*a*d)*(e*x)^(3/2)*(c + d*x
^2)^(5/2))/(221*d^2*e) + (2*b^2*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(17*d*e^3) - (8*c^(9/4)*(221*a^2*d^2 + 3*b*c*(7
*b*c - 34*a*d))*Sqrt[e]*(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])/(3315*d^(11/4)*Sqrt[c + d*x^2]) + (4*c^(9/4)*(221*a^2*d^2 + 3*b*c*(
7*b*c - 34*a*d))*Sqrt[e]*(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])/(3315*d^(11/4)*Sqrt[c + d*x^2])

Rule 464

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

Rule 459

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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && 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 \sqrt{e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac{2 \int \sqrt{e x} \left (c+d x^2\right )^{3/2} \left (\frac{17 a^2 d}{2}-\frac{1}{2} b (7 b c-34 a d) x^2\right ) \, dx}{17 d}\\ &=-\frac{2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}-\frac{1}{221} \left (-221 a^2-\frac{3 b c (7 b c-34 a d)}{d^2}\right ) \int \sqrt{e x} \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac{2 \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac{2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac{1}{663} \left (2 c \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right )\right ) \int \sqrt{e x} \sqrt{c+d x^2} \, dx\\ &=\frac{4 c \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \sqrt{c+d x^2}}{3315 e}+\frac{2 \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac{2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac{\left (4 c^2 \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right )\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{3315}\\ &=\frac{4 c \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \sqrt{c+d x^2}}{3315 e}+\frac{2 \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac{2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac{\left (8 c^2 \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3315 e}\\ &=\frac{4 c \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \sqrt{c+d x^2}}{3315 e}+\frac{2 \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac{2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac{\left (8 c^{5/2} \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3315 \sqrt{d}}-\frac{\left (8 c^{5/2} \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right )\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 )}{3315 \sqrt{d}}\\ &=\frac{4 c \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \sqrt{c+d x^2}}{3315 e}+\frac{8 c^2 \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) \sqrt{e x} \sqrt{c+d x^2}}{3315 \sqrt{d} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac{2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac{2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}-\frac{8 c^{9/4} \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) \sqrt{e} \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 )}{3315 d^{3/4} \sqrt{c+d x^2}}+\frac{4 c^{9/4} \left (221 a^2+\frac{3 b c (7 b c-34 a d)}{d^2}\right ) \sqrt{e} \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 )}{3315 d^{3/4} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.150643, size = 179, normalized size = 0.37 \[ \frac{2 \sqrt{e x} \left (12 c^2 x \sqrt{\frac{c}{d x^2}+1} \left (221 a^2 d^2-102 a b c d+21 b^2 c^2\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right )-x \left (c+d x^2\right ) \left (-221 a^2 d^2 \left (11 c+5 d x^2\right )-102 a b d \left (4 c^2+25 c d x^2+15 d^2 x^4\right )+b^2 \left (-60 c^2 d x^2+84 c^3-855 c d^2 x^4-585 d^3 x^6\right )\right )\right )}{9945 d^2 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*x]*(-(x*(c + d*x^2)*(-221*a^2*d^2*(11*c + 5*d*x^2) - 102*a*b*d*(4*c^2 + 25*c*d*x^2 + 15*d^2*x^4) + b
^2*(84*c^3 - 60*c^2*d*x^2 - 855*c*d^2*x^4 - 585*d^3*x^6))) + 12*c^2*(21*b^2*c^2 - 102*a*b*c*d + 221*a^2*d^2)*S
qrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(9945*d^2*Sqrt[c + d*x^2])

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Maple [A]  time = 0.017, size = 699, normalized size = 1.5 \begin{align*}{\frac{2}{9945\,{d}^{3}x}\sqrt{ex} \left ( 585\,{x}^{10}{b}^{2}{d}^{5}+1530\,{x}^{8}ab{d}^{5}+1440\,{x}^{8}{b}^{2}c{d}^{4}+1105\,{x}^{6}{a}^{2}{d}^{5}+4080\,{x}^{6}abc{d}^{4}+915\,{x}^{6}{b}^{2}{c}^{2}{d}^{3}+2652\,\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 ){a}^{2}{c}^{3}{d}^{2}-1224\,\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 ) ab{c}^{4}d+252\,\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 ){b}^{2}{c}^{5}-1326\,\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 ){a}^{2}{c}^{3}{d}^{2}+612\,\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 ) ab{c}^{4}d-126\,\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 ){b}^{2}{c}^{5}+3536\,{x}^{4}{a}^{2}c{d}^{4}+2958\,{x}^{4}ab{c}^{2}{d}^{3}-24\,{x}^{4}{b}^{2}{c}^{3}{d}^{2}+2431\,{x}^{2}{a}^{2}{c}^{2}{d}^{3}+408\,{x}^{2}ab{c}^{3}{d}^{2}-84\,{x}^{2}{b}^{2}{c}^{4}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^(3/2)*(e*x)^(1/2),x)

[Out]

2/9945/(d*x^2+c)^(1/2)*(e*x)^(1/2)/d^3*(585*x^10*b^2*d^5+1530*x^8*a*b*d^5+1440*x^8*b^2*c*d^4+1105*x^6*a^2*d^5+
4080*x^6*a*b*c*d^4+915*x^6*b^2*c^2*d^3+2652*((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)
)*a^2*c^3*d^2-1224*((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))*a*b*c^4*d+252*((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)*Elli
pticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^5-1326*((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))*a^2*c^3*d^2+612*((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))*a*b*c^4*d-126*((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))*b^2*c^5+3536*x^4*a^2*c*d
^4+2958*x^4*a*b*c^2*d^3-24*x^4*b^2*c^3*d^2+2431*x^2*a^2*c^2*d^3+408*x^2*a*b*c^3*d^2-84*x^2*b^2*c^4*d)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)*(e*x)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)*(e*x)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 26.8029, size = 304, normalized size = 0.63 \begin{align*} \frac{a^{2} c^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{a^{2} \sqrt{c} d \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{a b c^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{a b \sqrt{c} d \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{e^{5} \Gamma \left (\frac{15}{4}\right )} + \frac{b^{2} c^{\frac{3}{2}} \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{5} \Gamma \left (\frac{15}{4}\right )} + \frac{b^{2} \sqrt{c} d \left (e x\right )^{\frac{15}{2}} \Gamma \left (\frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{15}{4} \\ \frac{19}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{7} \Gamma \left (\frac{19}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)*(e*x)**(1/2),x)

[Out]

a**2*c**(3/2)*(e*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(2*e*gamma(7/4)) +
a**2*sqrt(c)*d*(e*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**3*gamma(11/
4)) + a*b*c**(3/2)*(e*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(e**3*gamma(1
1/4)) + a*b*sqrt(c)*d*(e*x)**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(e**5*g
amma(15/4)) + b**2*c**(3/2)*(e*x)**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(
2*e**5*gamma(15/4)) + b**2*sqrt(c)*d*(e*x)**(15/2)*gamma(15/4)*hyper((-1/2, 15/4), (19/4,), d*x**2*exp_polar(I
*pi)/c)/(2*e**7*gamma(19/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)*(e*x)^(1/2),x, algorithm="giac")

[Out]

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