3.140 \(\int \frac{1}{\left (a-b x^2\right )^{7/3} \left (3 a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=827 \[ \frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (b x^2+3 a\right )}+\frac{79 x}{768 a^4 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{79 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{512\ 3^{3/4} a^{11/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac{79 \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{384 \sqrt{2} \sqrt [4]{3} a^{11/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

[Out]

(5*x)/(384*a^3*(a - b*x^2)^(4/3)) + (79*x)/(768*a^4*(a - b*x^2)^(1/3)) + x/(24*a
^2*(a - b*x^2)^(4/3)*(3*a + b*x^2)) + (79*x)/(768*a^4*((1 - Sqrt[3])*a^(1/3) - (
a - b*x^2)^(1/3))) + (Sqrt[3]*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(128*2^(2/3
)*a^(23/6)*Sqrt[b]) + (Sqrt[3]*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b
*x^2)^(1/3)))/(Sqrt[b]*x)])/(128*2^(2/3)*a^(23/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)
/Sqrt[a]]/(128*2^(2/3)*a^(23/6)*Sqrt[b]) + (3*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1
/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(128*2^(2/3)*a^(23/6)*Sqrt[b]) + (79*Sqrt[2
+ Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/
3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE
[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a
- b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(512*3^(3/4)*a^(11/3)*b*x*Sqrt[-((a^(1/3)*(a^
(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (
79*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a
- b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[
((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)
^(1/3))], -7 + 4*Sqrt[3]])/(384*Sqrt[2]*3^(1/4)*a^(11/3)*b*x*Sqrt[-((a^(1/3)*(a^
(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

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Rubi [A]  time = 1.52737, antiderivative size = 827, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (b x^2+3 a\right )}+\frac{79 x}{768 a^4 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{79 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{512\ 3^{3/4} a^{11/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac{79 \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{384 \sqrt{2} \sqrt [4]{3} a^{11/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)^2),x]

[Out]

(5*x)/(384*a^3*(a - b*x^2)^(4/3)) + (79*x)/(768*a^4*(a - b*x^2)^(1/3)) + x/(24*a
^2*(a - b*x^2)^(4/3)*(3*a + b*x^2)) + (79*x)/(768*a^4*((1 - Sqrt[3])*a^(1/3) - (
a - b*x^2)^(1/3))) + (Sqrt[3]*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(128*2^(2/3
)*a^(23/6)*Sqrt[b]) + (Sqrt[3]*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b
*x^2)^(1/3)))/(Sqrt[b]*x)])/(128*2^(2/3)*a^(23/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)
/Sqrt[a]]/(128*2^(2/3)*a^(23/6)*Sqrt[b]) + (3*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1
/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(128*2^(2/3)*a^(23/6)*Sqrt[b]) + (79*Sqrt[2
+ Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/
3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE
[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a
- b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(512*3^(3/4)*a^(11/3)*b*x*Sqrt[-((a^(1/3)*(a^
(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (
79*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a
- b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[
((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)
^(1/3))], -7 + 4*Sqrt[3]])/(384*Sqrt[2]*3^(1/4)*a^(11/3)*b*x*Sqrt[-((a^(1/3)*(a^
(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-b*x**2+a)**(7/3)/(b*x**2+3*a)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 0.427194, size = 346, normalized size = 0.42 \[ \frac{x \left (-\frac{1161 a^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}+\frac{897 a^2-444 a b x^2-237 b^2 x^4}{a-b x^2}-\frac{395 a b x^2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+15 a F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}\right )}{2304 a^4 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)^2),x]

[Out]

(x*((897*a^2 - 444*a*b*x^2 - 237*b^2*x^4)/(a - b*x^2) - (1161*a^2*AppellF1[1/2,
1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)
/a, -(b*x^2)/(3*a)] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -(b*x^2)/(
3*a)] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)])) - (395*a*b*x^2*A
ppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)])/(15*a*AppellF1[3/2, 1/3, 1
, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)] + 2*b*x^2*(-AppellF1[5/2, 1/3, 2, 7/2, (b*x^2)
/a, -(b*x^2)/(3*a)] + AppellF1[5/2, 4/3, 1, 7/2, (b*x^2)/a, -(b*x^2)/(3*a)]))))/
(2304*a^4*(a - b*x^2)^(1/3)*(3*a + b*x^2))

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( b{x}^{2}+3\,a \right ) ^{2}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{7}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^2+a)^(7/3)/(b*x^2+3*a)^2,x)

[Out]

int(1/(-b*x^2+a)^(7/3)/(b*x^2+3*a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}^{2}{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(7/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(7/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(7/3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x**2+a)**(7/3)/(b*x**2+3*a)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}^{2}{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(7/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + 3*a)^2*(-b*x^2 + a)^(7/3)), x)