3.66 \(\int \frac{x^6}{\left (a x+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=279 \[ -\frac{21 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a x+b x^3}}+\frac{21 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{a x+b x^3}}-\frac{21 a x \left (a+b x^2\right )}{5 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{7 x \sqrt{a x+b x^3}}{5 b^2}-\frac{x^4}{b \sqrt{a x+b x^3}} \]

[Out]

-(x^4/(b*Sqrt[a*x + b*x^3])) - (21*a*x*(a + b*x^2))/(5*b^(5/2)*(Sqrt[a] + Sqrt[b
]*x)*Sqrt[a*x + b*x^3]) + (7*x*Sqrt[a*x + b*x^3])/(5*b^2) + (21*a^(5/4)*Sqrt[x]*
(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcT
an[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*b^(11/4)*Sqrt[a*x + b*x^3]) - (21*a^(5/4
)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Ellipt
icF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(10*b^(11/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.533806, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{21 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a x+b x^3}}+\frac{21 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{a x+b x^3}}-\frac{21 a x \left (a+b x^2\right )}{5 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{7 x \sqrt{a x+b x^3}}{5 b^2}-\frac{x^4}{b \sqrt{a x+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a*x + b*x^3)^(3/2),x]

[Out]

-(x^4/(b*Sqrt[a*x + b*x^3])) - (21*a*x*(a + b*x^2))/(5*b^(5/2)*(Sqrt[a] + Sqrt[b
]*x)*Sqrt[a*x + b*x^3]) + (7*x*Sqrt[a*x + b*x^3])/(5*b^2) + (21*a^(5/4)*Sqrt[x]*
(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcT
an[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*b^(11/4)*Sqrt[a*x + b*x^3]) - (21*a^(5/4
)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Ellipt
icF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(10*b^(11/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 51.415, size = 260, normalized size = 0.93 \[ \frac{21 a^{\frac{5}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{11}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{21 a^{\frac{5}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{10 b^{\frac{11}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{21 a \sqrt{a x + b x^{3}}}{5 b^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{x^{4}}{b \sqrt{a x + b x^{3}}} + \frac{7 x \sqrt{a x + b x^{3}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b*x**3+a*x)**(3/2),x)

[Out]

21*a**(5/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*sq
rt(a*x + b*x**3)*elliptic_e(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(5*b**(11/4)
*sqrt(x)*(a + b*x**2)) - 21*a**(5/4)*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)*sqrt(x)/a**
(1/4)), 1/2)/(10*b**(11/4)*sqrt(x)*(a + b*x**2)) - 21*a*sqrt(a*x + b*x**3)/(5*b*
*(5/2)*(sqrt(a) + sqrt(b)*x)) - x**4/(b*sqrt(a*x + b*x**3)) + 7*x*sqrt(a*x + b*x
**3)/(5*b**2)

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Mathematica [C]  time = 0.178064, size = 173, normalized size = 0.62 \[ \frac{x \left (21 a^{3/2} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-21 a^{3/2} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (7 a+2 b x^2\right )\right )}{5 b^{5/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a*x + b*x^3)^(3/2),x]

[Out]

(x*(Sqrt[b]*x*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(7*a + 2*b*x^2) - 21*a^(3/2)*Sqrt[1 +
(b*x^2)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1] + 21*a^(3/2)*Sq
rt[1 + (b*x^2)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1]))/(5*b^(
5/2)*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*Sqrt[x*(a + b*x^2)])

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Maple [A]  time = 0.028, size = 200, normalized size = 0.7 \[{\frac{a{x}^{2}}{{b}^{2}}{\frac{1}{\sqrt{ \left ({x}^{2}+{\frac{a}{b}} \right ) xb}}}}+{\frac{2\,x}{5\,{b}^{2}}\sqrt{b{x}^{3}+ax}}-{\frac{21\,a}{10\,{b}^{3}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^3+a*x)^(3/2),x)

[Out]

x^2/b^2*a/((x^2+a/b)*x*b)^(1/2)+2/5*x*(b*x^3+a*x)^(1/2)/b^2-21/10*a/b^3*(-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)*(-2/b*(-a*b)^(1/2)*E
llipticE(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/b*(-a*b)^(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^3 + a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{5}}{\sqrt{b x^{3} + a x}{\left (b x^{2} + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^3 + a*x)^(3/2),x, algorithm="fricas")

[Out]

integral(x^5/(sqrt(b*x^3 + a*x)*(b*x^2 + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**3+a*x)**(3/2),x)

[Out]

Integral(x**6/(x*(a + b*x**2))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^3 + a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^6/(b*x^3 + a*x)^(3/2), x)