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

Optimal. Leaf size=288 \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}-\frac{2 \sqrt{e x} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 c d e^3}-\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 d e^3}-\frac{4 c^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{5/4} e^{5/2} \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3} \]

[Out]

(-4*(3*b^2*c^2 - 11*a*d*(6*b*c + 7*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d*e^3)
- (2*(3*b^2*c^2 - 11*a*d*(6*b*c + 7*a*d))*Sqrt[e*x]*(c + d*x^2)^(3/2))/(231*c*d*
e^3) - (2*a^2*(c + d*x^2)^(5/2))/(3*c*e*(e*x)^(3/2)) + (2*b^2*Sqrt[e*x]*(c + d*x
^2)^(5/2))/(11*d*e^3) - (4*c^(3/4)*(3*b^2*c^2 - 11*a*d*(6*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])/(231*d^(5/4)*e^(5/2)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.622613, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}-\frac{2 \sqrt{e x} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 c d e^3}-\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 d e^3}-\frac{4 c^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{5/4} e^{5/2} \sqrt{c+d x^2}}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{5/2}}{11 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(3*b^2*c^2 - 11*a*d*(6*b*c + 7*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d*e^3)
- (2*(3*b^2*c^2 - 11*a*d*(6*b*c + 7*a*d))*Sqrt[e*x]*(c + d*x^2)^(3/2))/(231*c*d*
e^3) - (2*a^2*(c + d*x^2)^(5/2))/(3*c*e*(e*x)^(3/2)) + (2*b^2*Sqrt[e*x]*(c + d*x
^2)^(5/2))/(11*d*e^3) - (4*c^(3/4)*(3*b^2*c^2 - 11*a*d*(6*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])/(231*d^(5/4)*e^(5/2)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 59.478, size = 272, normalized size = 0.94 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{3 c e \left (e x\right )^{\frac{3}{2}}} + \frac{2 b^{2} \sqrt{e x} \left (c + d x^{2}\right )^{\frac{5}{2}}}{11 d e^{3}} - \frac{4 c^{\frac{3}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- 11 a d \left (7 a d + 6 b c\right ) + 3 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{231 d^{\frac{5}{4}} e^{\frac{5}{2}} \sqrt{c + d x^{2}}} - \frac{4 \sqrt{e x} \sqrt{c + d x^{2}} \left (- 11 a d \left (7 a d + 6 b c\right ) + 3 b^{2} c^{2}\right )}{231 d e^{3}} - \frac{2 \sqrt{e x} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (- 11 a d \left (7 a d + 6 b c\right ) + 3 b^{2} c^{2}\right )}{231 c d e^{3}} \]

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)**(5/2),x)

[Out]

-2*a**2*(c + d*x**2)**(5/2)/(3*c*e*(e*x)**(3/2)) + 2*b**2*sqrt(e*x)*(c + d*x**2)
**(5/2)/(11*d*e**3) - 4*c**(3/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sq
rt(c) + sqrt(d)*x)*(-11*a*d*(7*a*d + 6*b*c) + 3*b**2*c**2)*elliptic_f(2*atan(d**
(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(231*d**(5/4)*e**(5/2)*sqrt(c + d*x**2
)) - 4*sqrt(e*x)*sqrt(c + d*x**2)*(-11*a*d*(7*a*d + 6*b*c) + 3*b**2*c**2)/(231*d
*e**3) - 2*sqrt(e*x)*(c + d*x**2)**(3/2)*(-11*a*d*(7*a*d + 6*b*c) + 3*b**2*c**2)
/(231*c*d*e**3)

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Mathematica [C]  time = 0.366136, size = 202, normalized size = 0.7 \[ \frac{x^{5/2} \left (\frac{2 \left (c+d x^2\right ) \left (77 a^2 d \left (d x^2-c\right )+66 a b d x^2 \left (3 c+d x^2\right )+3 b^2 x^2 \left (4 c^2+13 c d x^2+7 d^2 x^4\right )\right )}{d x^{3/2}}+\frac{8 i c x \sqrt{\frac{c}{d x^2}+1} \left (77 a^2 d^2+66 a b c d-3 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{231 (e x)^{5/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(5/2)*((2*(c + d*x^2)*(77*a^2*d*(-c + d*x^2) + 66*a*b*d*x^2*(3*c + d*x^2) + 3
*b^2*x^2*(4*c^2 + 13*c*d*x^2 + 7*d^2*x^4)))/(d*x^(3/2)) + ((8*I)*c*(-3*b^2*c^2 +
66*a*b*c*d + 77*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt
[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d)))/(231*(e*x)^(5/2)*Sq
rt[c + d*x^2])

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Maple [A]  time = 0.028, size = 415, normalized size = 1.4 \[{\frac{2}{231\,x{d}^{2}{e}^{2}} \left ( 21\,{x}^{8}{b}^{2}{d}^{4}+154\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) x{a}^{2}c{d}^{2}+132\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) xab{c}^{2}d-6\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) x{b}^{2}{c}^{3}+66\,{x}^{6}ab{d}^{4}+60\,{x}^{6}{b}^{2}c{d}^{3}+77\,{x}^{4}{a}^{2}{d}^{4}+264\,{x}^{4}abc{d}^{3}+51\,{x}^{4}{b}^{2}{c}^{2}{d}^{2}+198\,{x}^{2}ab{c}^{2}{d}^{2}+12\,{x}^{2}{b}^{2}{c}^{3}d-77\,{a}^{2}{c}^{2}{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

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)^(5/2),x)

[Out]

2/231/(d*x^2+c)^(1/2)/x*(21*x^8*b^2*d^4+154*(-c*d)^(1/2)*((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*a^2
*c*d^2+132*(-c*d)^(1/2)*((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*a*b*c^2*d-6*(-c*d)^(1/2)*((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*b^2*c^3+66*x^6*a*b*d^4+60*x^6*b^2*c*d^3+77*x^4*a^2*d^4+264*x^4*a*b*c*d^
3+51*x^4*b^2*c^2*d^2+198*x^2*a*b*c^2*d^2+12*x^2*b^2*c^3*d-77*a^2*c^2*d^2)/d^2/e^
2/(e*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]

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)^(5/2),x, algorithm="maxima")

[Out]

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

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

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)^(5/2),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^2*x^2), 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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]

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)^(5/2),x, algorithm="giac")

[Out]

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