3.56 \(\int \frac{\left (a x+b x^3\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=163 \[ -\frac{4 b^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 a^{5/4} \sqrt{a x+b x^3}}-\frac{8 b^2 \sqrt{a x+b x^3}}{77 a x^2}-\frac{2 \left (a x+b x^3\right )^{3/2}}{11 x^7}-\frac{12 b \sqrt{a x+b x^3}}{77 x^4} \]

[Out]

(-12*b*Sqrt[a*x + b*x^3])/(77*x^4) - (8*b^2*Sqrt[a*x + b*x^3])/(77*a*x^2) - (2*(
a*x + b*x^3)^(3/2))/(11*x^7) - (4*b^(11/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a
+ b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)]
, 1/2])/(77*a^(5/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.337996, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{4 b^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 a^{5/4} \sqrt{a x+b x^3}}-\frac{8 b^2 \sqrt{a x+b x^3}}{77 a x^2}-\frac{2 \left (a x+b x^3\right )^{3/2}}{11 x^7}-\frac{12 b \sqrt{a x+b x^3}}{77 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^3)^(3/2)/x^8,x]

[Out]

(-12*b*Sqrt[a*x + b*x^3])/(77*x^4) - (8*b^2*Sqrt[a*x + b*x^3])/(77*a*x^2) - (2*(
a*x + b*x^3)^(3/2))/(11*x^7) - (4*b^(11/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a
+ b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)]
, 1/2])/(77*a^(5/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 31.3691, size = 158, normalized size = 0.97 \[ - \frac{12 b \sqrt{a x + b x^{3}}}{77 x^{4}} - \frac{2 \left (a x + b x^{3}\right )^{\frac{3}{2}}}{11 x^{7}} - \frac{8 b^{2} \sqrt{a x + b x^{3}}}{77 a x^{2}} - \frac{4 b^{\frac{11}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{77 a^{\frac{5}{4}} \sqrt{x} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a*x)**(3/2)/x**8,x)

[Out]

-12*b*sqrt(a*x + b*x**3)/(77*x**4) - 2*(a*x + b*x**3)**(3/2)/(11*x**7) - 8*b**2*
sqrt(a*x + b*x**3)/(77*a*x**2) - 4*b**(11/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b
)*x)**2)*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x**3)*elliptic_f(2*atan(b**(1/4)*sqr
t(x)/a**(1/4)), 1/2)/(77*a**(5/4)*sqrt(x)*(a + b*x**2))

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Mathematica [C]  time = 0.297343, size = 150, normalized size = 0.92 \[ -\frac{2 \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (7 a^3+20 a^2 b x^2+17 a b^2 x^4+4 b^3 x^6\right )+4 i b^3 x^{13/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{77 a x^5 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^3)^(3/2)/x^8,x]

[Out]

(-2*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(7*a^3 + 20*a^2*b*x^2 + 17*a*b^2*x^4 + 4*b^3*x^6)
+ (4*I)*b^3*Sqrt[1 + a/(b*x^2)]*x^(13/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/S
qrt[b]]/Sqrt[x]], -1]))/(77*a*Sqrt[(I*Sqrt[a])/Sqrt[b]]*x^5*Sqrt[x*(a + b*x^2)])

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Maple [A]  time = 0.031, size = 169, normalized size = 1. \[ -{\frac{2\,a}{11\,{x}^{6}}\sqrt{b{x}^{3}+ax}}-{\frac{26\,b}{77\,{x}^{4}}\sqrt{b{x}^{3}+ax}}-{\frac{8\,{b}^{2}}{77\,a{x}^{2}}\sqrt{b{x}^{3}+ax}}-{\frac{4\,{b}^{2}}{77\,a}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x)^(3/2)/x^8,x)

[Out]

-2/11*a*(b*x^3+a*x)^(1/2)/x^6-26/77*b*(b*x^3+a*x)^(1/2)/x^4-8/77*b^2*(b*x^3+a*x)
^(1/2)/a/x^2-4/77/a*b^2*(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)
*(-2*(x-1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)/(b*x^3
+a*x)^(1/2)*EllipticF(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x)^(3/2)/x^8,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a*x)^(3/2)/x^8, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{3} + a x}{\left (b x^{2} + a\right )}}{x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x)^(3/2)/x^8,x, algorithm="fricas")

[Out]

integral(sqrt(b*x^3 + a*x)*(b*x^2 + a)/x^7, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x)**(3/2)/x**8,x)

[Out]

Integral((x*(a + b*x**2))**(3/2)/x**8, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x)^(3/2)/x^8,x, algorithm="giac")

[Out]

integrate((b*x^3 + a*x)^(3/2)/x^8, x)