3.848 \(\int \frac{\left (a+b x^2\right )^2}{(e x)^{13/2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=242 \[ -\frac{2 a^2 \sqrt{c+d x^2}}{11 c e (e x)^{11/2}}-\frac{d^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 c^{13/4} e^{13/2} \sqrt{c+d x^2}}-\frac{2 \sqrt{c+d x^2} \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right )}{231 c^3 e^5 (e x)^{3/2}}-\frac{2 a \sqrt{c+d x^2} (22 b c-9 a d)}{77 c^2 e^3 (e x)^{7/2}} \]

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(11*c*e*(e*x)^(11/2)) - (2*a*(22*b*c - 9*a*d)*Sqrt[c +
d*x^2])/(77*c^2*e^3*(e*x)^(7/2)) - (2*(77*b^2*c^2 - 5*a*d*(22*b*c - 9*a*d))*Sqrt
[c + d*x^2])/(231*c^3*e^5*(e*x)^(3/2)) - (d^(3/4)*(77*b^2*c^2 - 5*a*d*(22*b*c -
9*a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Elliptic
F[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(231*c^(13/4)*e^(13/2)*
Sqrt[c + d*x^2])

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Rubi [A]  time = 0.573652, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 a^2 \sqrt{c+d x^2}}{11 c e (e x)^{11/2}}-\frac{d^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 c^{13/4} e^{13/2} \sqrt{c+d x^2}}-\frac{2 \sqrt{c+d x^2} \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right )}{231 c^3 e^5 (e x)^{3/2}}-\frac{2 a \sqrt{c+d x^2} (22 b c-9 a d)}{77 c^2 e^3 (e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/((e*x)^(13/2)*Sqrt[c + d*x^2]),x]

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(11*c*e*(e*x)^(11/2)) - (2*a*(22*b*c - 9*a*d)*Sqrt[c +
d*x^2])/(77*c^2*e^3*(e*x)^(7/2)) - (2*(77*b^2*c^2 - 5*a*d*(22*b*c - 9*a*d))*Sqrt
[c + d*x^2])/(231*c^3*e^5*(e*x)^(3/2)) - (d^(3/4)*(77*b^2*c^2 - 5*a*d*(22*b*c -
9*a*d))*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Elliptic
F[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(231*c^(13/4)*e^(13/2)*
Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 52.4674, size = 231, normalized size = 0.95 \[ - \frac{2 a^{2} \sqrt{c + d x^{2}}}{11 c e \left (e x\right )^{\frac{11}{2}}} + \frac{2 a \sqrt{c + d x^{2}} \left (9 a d - 22 b c\right )}{77 c^{2} e^{3} \left (e x\right )^{\frac{7}{2}}} - \frac{2 \sqrt{c + d x^{2}} \left (5 a d \left (9 a d - 22 b c\right ) + 77 b^{2} c^{2}\right )}{231 c^{3} e^{5} \left (e x\right )^{\frac{3}{2}}} - \frac{d^{\frac{3}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (5 a d \left (9 a d - 22 b c\right ) + 77 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{231 c^{\frac{13}{4}} e^{\frac{13}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/(e*x)**(13/2)/(d*x**2+c)**(1/2),x)

[Out]

-2*a**2*sqrt(c + d*x**2)/(11*c*e*(e*x)**(11/2)) + 2*a*sqrt(c + d*x**2)*(9*a*d -
22*b*c)/(77*c**2*e**3*(e*x)**(7/2)) - 2*sqrt(c + d*x**2)*(5*a*d*(9*a*d - 22*b*c)
+ 77*b**2*c**2)/(231*c**3*e**5*(e*x)**(3/2)) - d**(3/4)*sqrt((c + d*x**2)/(sqrt
(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(5*a*d*(9*a*d - 22*b*c) + 77*b**2*c**
2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(231*c**(13/4)
*e**(13/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.376315, size = 196, normalized size = 0.81 \[ \frac{x^{13/2} \left (-\frac{2 \left (c+d x^2\right ) \left (3 a^2 \left (7 c^2-9 c d x^2+15 d^2 x^4\right )+22 a b c x^2 \left (3 c-5 d x^2\right )+77 b^2 c^2 x^4\right )}{c^3 x^{11/2}}-\frac{2 i d x \sqrt{\frac{c}{d x^2}+1} \left (45 a^2 d^2-110 a b c d+77 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{c^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{231 (e x)^{13/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/((e*x)^(13/2)*Sqrt[c + d*x^2]),x]

[Out]

(x^(13/2)*((-2*(c + d*x^2)*(77*b^2*c^2*x^4 + 22*a*b*c*x^2*(3*c - 5*d*x^2) + 3*a^
2*(7*c^2 - 9*c*d*x^2 + 15*d^2*x^4)))/(c^3*x^(11/2)) - ((2*I)*d*(77*b^2*c^2 - 110
*a*b*c*d + 45*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]
)/Sqrt[d]]/Sqrt[x]], -1])/(c^3*Sqrt[(I*Sqrt[c])/Sqrt[d]])))/(231*(e*x)^(13/2)*Sq
rt[c + d*x^2])

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Maple [A]  time = 0.051, size = 411, normalized size = 1.7 \[ -{\frac{1}{231\,{x}^{5}{c}^{3}{e}^{6}} \left ( 45\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}{a}^{2}{d}^{2}-110\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}abcd+77\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}{b}^{2}{c}^{2}+90\,{x}^{6}{a}^{2}{d}^{3}-220\,{x}^{6}abc{d}^{2}+154\,{x}^{6}{b}^{2}{c}^{2}d+36\,{x}^{4}{a}^{2}c{d}^{2}-88\,{x}^{4}ab{c}^{2}d+154\,{x}^{4}{b}^{2}{c}^{3}-12\,{x}^{2}{a}^{2}{c}^{2}d+132\,{x}^{2}ab{c}^{3}+42\,{a}^{2}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(e*x)^(13/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/231/(d*x^2+c)^(1/2)/x^5*(45*((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))*(-c*d)^(1/2)*x^5*a^2*d^2-110*((
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))*(-c*d)^(1/2)*x^5*a*b*c*d+77*((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)*E
llipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^5*b
^2*c^2+90*x^6*a^2*d^3-220*x^6*a*b*c*d^2+154*x^6*b^2*c^2*d+36*x^4*a^2*c*d^2-88*x^
4*a*b*c^2*d+154*x^4*b^2*c^3-12*x^2*a^2*c^2*d+132*x^2*a*b*c^3+42*a^2*c^3)/c^3/e^6
/(e*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{\sqrt{d x^{2} + c} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(13/2)),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(13/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(13/2)),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)/(sqrt(d*x^2 + c)*sqrt(e*x)*e^6*x^6), 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/(e*x)**(13/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{\sqrt{d x^{2} + c} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(13/2)),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(13/2)), x)