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

Optimal. Leaf size=468 \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}+\frac{4 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{8 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{9 c^2 e^5}+\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 c e^5}+\frac{8 \sqrt{e x} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a \left (c+d x^2\right )^{5/2} (a d+2 b c)}{c^2 e^3 \sqrt{e x}} \]

[Out]

(4*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(15*c*e^5) + (8*
(b^2*c^2 + 9*a*d*(2*b*c + a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*Sqrt[d]*e^4*(Sqrt
[c] + Sqrt[d]*x)) + (2*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(e*x)^(3/2)*(c + d*x^2)^(
3/2))/(9*c^2*e^5) - (2*a^2*(c + d*x^2)^(5/2))/(5*c*e*(e*x)^(5/2)) - (2*a*(2*b*c
+ a*d)*(c + d*x^2)^(5/2))/(c^2*e^3*Sqrt[e*x]) - (8*c^(1/4)*(b^2*c^2 + 9*a*d*(2*b
*c + a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Ellip
ticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(15*d^(3/4)*e^(7/2)*
Sqrt[c + d*x^2]) + (4*c^(1/4)*(b^2*c^2 + 9*a*d*(2*b*c + 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])/(15*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2])

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Rubi [A]  time = 1.07944, antiderivative size = 468, 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 \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{5 c e (e x)^{5/2}}+\frac{4 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{8 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (9 a d (a d+2 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{9 c^2 e^5}+\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 c e^5}+\frac{8 \sqrt{e x} \sqrt{c+d x^2} \left (9 a d (a d+2 b c)+b^2 c^2\right )}{15 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a \left (c+d x^2\right )^{5/2} (a d+2 b c)}{c^2 e^3 \sqrt{e x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(15*c*e^5) + (8*
(b^2*c^2 + 9*a*d*(2*b*c + a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*Sqrt[d]*e^4*(Sqrt
[c] + Sqrt[d]*x)) + (2*(b^2*c^2 + 9*a*d*(2*b*c + a*d))*(e*x)^(3/2)*(c + d*x^2)^(
3/2))/(9*c^2*e^5) - (2*a^2*(c + d*x^2)^(5/2))/(5*c*e*(e*x)^(5/2)) - (2*a*(2*b*c
+ a*d)*(c + d*x^2)^(5/2))/(c^2*e^3*Sqrt[e*x]) - (8*c^(1/4)*(b^2*c^2 + 9*a*d*(2*b
*c + a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Ellip
ticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(15*d^(3/4)*e^(7/2)*
Sqrt[c + d*x^2]) + (4*c^(1/4)*(b^2*c^2 + 9*a*d*(2*b*c + 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])/(15*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 105.925, size = 444, normalized size = 0.95 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 c e \left (e x\right )^{\frac{5}{2}}} - \frac{2 a \left (c + d x^{2}\right )^{\frac{5}{2}} \left (a d + 2 b c\right )}{c^{2} e^{3} \sqrt{e x}} - \frac{8 \sqrt [4]{c} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} + \frac{4 \sqrt [4]{c} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} + \frac{8 \sqrt{e x} \sqrt{c + d x^{2}} \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right )}{15 \sqrt{d} e^{4} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{4 \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right )}{15 c e^{5}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (9 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right )}{9 c^{2} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*(c + d*x**2)**(5/2)/(5*c*e*(e*x)**(5/2)) - 2*a*(c + d*x**2)**(5/2)*(a*d
+ 2*b*c)/(c**2*e**3*sqrt(e*x)) - 8*c**(1/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)
*x)**2)*(sqrt(c) + sqrt(d)*x)*(9*a*d*(a*d + 2*b*c) + b**2*c**2)*elliptic_e(2*ata
n(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(15*d**(3/4)*e**(7/2)*sqrt(c + d*
x**2)) + 4*c**(1/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(
d)*x)*(9*a*d*(a*d + 2*b*c) + b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c*
*(1/4)*sqrt(e))), 1/2)/(15*d**(3/4)*e**(7/2)*sqrt(c + d*x**2)) + 8*sqrt(e*x)*sqr
t(c + d*x**2)*(9*a*d*(a*d + 2*b*c) + b**2*c**2)/(15*sqrt(d)*e**4*(sqrt(c) + sqrt
(d)*x)) + 4*(e*x)**(3/2)*sqrt(c + d*x**2)*(9*a*d*(a*d + 2*b*c) + b**2*c**2)/(15*
c*e**5) + 2*(e*x)**(3/2)*(c + d*x**2)**(3/2)*(9*a*d*(a*d + 2*b*c) + b**2*c**2)/(
9*c**2*e**5)

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Mathematica [C]  time = 1.51636, size = 240, normalized size = 0.51 \[ \frac{x^{7/2} \left (\frac{2 \sqrt{c+d x^2} \left (-9 a^2 \left (c+7 d x^2\right )+18 a b x^2 \left (d x^2-5 c\right )+b^2 x^4 \left (11 c+5 d x^2\right )\right )}{3 x^{5/2}}-\frac{8 x \left (9 a^2 d^2+18 a b c d+b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )}{d \sqrt{c+d x^2}}\right )}{15 (e x)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(7/2)*((2*Sqrt[c + d*x^2]*(18*a*b*x^2*(-5*c + d*x^2) + b^2*x^4*(11*c + 5*d*x^
2) - 9*a^2*(c + 7*d*x^2)))/(3*x^(5/2)) - (8*(b^2*c^2 + 18*a*b*c*d + 9*a^2*d^2)*x
*(-((d + c/x^2)*Sqrt[x]) + (I*c*Sqrt[1 + c/(d*x^2)]*(EllipticE[I*ArcSinh[Sqrt[(I
*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]
/Sqrt[x]], -1]))/((I*Sqrt[c])/Sqrt[d])^(3/2)))/(d*Sqrt[c + d*x^2])))/(15*(e*x)^(
7/2))

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Maple [A]  time = 0.031, size = 668, normalized size = 1.4 \[{\frac{2}{45\,d{x}^{2}{e}^{3}} \left ( 5\,{b}^{2}{d}^{3}{x}^{8}+108\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}+216\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d+12\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-54\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}-108\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d-6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}+18\,{x}^{6}ab{d}^{3}+16\,{x}^{6}{b}^{2}c{d}^{2}-63\,{x}^{4}{a}^{2}{d}^{3}-72\,{x}^{4}abc{d}^{2}+11\,{x}^{4}{b}^{2}{c}^{2}d-72\,{x}^{2}{a}^{2}c{d}^{2}-90\,{x}^{2}ab{c}^{2}d-9\,{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

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)^(7/2),x)

[Out]

2/45/x^2*(5*b^2*d^3*x^8+108*((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+216*((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*b*c^2*d+12*((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^3-54*((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^2-108*((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-6*((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)*Ell
ipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^2*c^3+18*x^6*a
*b*d^3+16*x^6*b^2*c*d^2-63*x^4*a^2*d^3-72*x^4*a*b*c*d^2+11*x^4*b^2*c^2*d-72*x^2*
a^2*c*d^2-90*x^2*a*b*c^2*d-9*a^2*c^2*d)/(d*x^2+c)^(1/2)/d/e^3/(e*x)^(1/2)

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

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)^(7/2),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. \[{\rm integral}\left (\frac{{\left (b^{2} d x^{6} +{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c +{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{\sqrt{e x} e^{3} x^{3}}, x\right ) \]

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)^(7/2),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)*sqr
t(d*x^2 + c)/(sqrt(e*x)*e^3*x^3), x)

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

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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]

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)^(7/2),x, algorithm="giac")

[Out]

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