3.154 \(\int \frac{x^6 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{2 \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}+\frac{x \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac{x^5 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

[Out]

-(((b*B - A*c)*x^5)/(b*c*Sqrt[b*x^2 + c*x^4])) - (2*(4*b*B - 3*A*c)*Sqrt[b*x^2 +
c*x^4])/(3*c^3*x) + ((4*b*B - 3*A*c)*x*Sqrt[b*x^2 + c*x^4])/(3*b*c^2)

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Rubi [A]  time = 0.305264, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{2 \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}+\frac{x \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac{x^5 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(((b*B - A*c)*x^5)/(b*c*Sqrt[b*x^2 + c*x^4])) - (2*(4*b*B - 3*A*c)*Sqrt[b*x^2 +
c*x^4])/(3*c^3*x) + ((4*b*B - 3*A*c)*x*Sqrt[b*x^2 + c*x^4])/(3*b*c^2)

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Rubi in Sympy [A]  time = 29.418, size = 90, normalized size = 0.87 \[ \frac{2 \left (3 A c - 4 B b\right ) \sqrt{b x^{2} + c x^{4}}}{3 c^{3} x} + \frac{x^{5} \left (A c - B b\right )}{b c \sqrt{b x^{2} + c x^{4}}} - \frac{x \left (3 A c - 4 B b\right ) \sqrt{b x^{2} + c x^{4}}}{3 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(B*x**2+A)/(c*x**4+b*x**2)**(3/2),x)

[Out]

2*(3*A*c - 4*B*b)*sqrt(b*x**2 + c*x**4)/(3*c**3*x) + x**5*(A*c - B*b)/(b*c*sqrt(
b*x**2 + c*x**4)) - x*(3*A*c - 4*B*b)*sqrt(b*x**2 + c*x**4)/(3*b*c**2)

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Mathematica [A]  time = 0.0688603, size = 60, normalized size = 0.58 \[ \frac{x \left (b \left (6 A c-4 B c x^2\right )+c^2 x^2 \left (3 A+B x^2\right )-8 b^2 B\right )}{3 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*(-8*b^2*B + c^2*x^2*(3*A + B*x^2) + b*(6*A*c - 4*B*c*x^2)))/(3*c^3*Sqrt[x^2*(
b + c*x^2)])

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Maple [A]  time = 0.007, size = 66, normalized size = 0.6 \[{\frac{ \left ( c{x}^{2}+b \right ) \left ( B{c}^{2}{x}^{4}+3\,A{x}^{2}{c}^{2}-4\,B{x}^{2}bc+6\,Abc-8\,{b}^{2}B \right ){x}^{3}}{3\,{c}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x)

[Out]

1/3*(c*x^2+b)*(B*c^2*x^4+3*A*c^2*x^2-4*B*b*c*x^2+6*A*b*c-8*B*b^2)*x^3/c^3/(c*x^4
+b*x^2)^(3/2)

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Maxima [A]  time = 1.3893, size = 80, normalized size = 0.77 \[ \frac{{\left (c x^{2} + 2 \, b\right )} A}{\sqrt{c x^{2} + b} c^{2}} + \frac{{\left (c^{2} x^{4} - 4 \, b c x^{2} - 8 \, b^{2}\right )} B}{3 \, \sqrt{c x^{2} + b} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^6/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")

[Out]

(c*x^2 + 2*b)*A/(sqrt(c*x^2 + b)*c^2) + 1/3*(c^2*x^4 - 4*b*c*x^2 - 8*b^2)*B/(sqr
t(c*x^2 + b)*c^3)

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Fricas [A]  time = 0.225242, size = 92, normalized size = 0.88 \[ \frac{{\left (B c^{2} x^{4} - 8 \, B b^{2} + 6 \, A b c -{\left (4 \, B b c - 3 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3 \,{\left (c^{4} x^{3} + b c^{3} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^6/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(B*c^2*x^4 - 8*B*b^2 + 6*A*b*c - (4*B*b*c - 3*A*c^2)*x^2)*sqrt(c*x^4 + b*x^2
)/(c^4*x^3 + b*c^3*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(B*x**2+A)/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x**6*(A + B*x**2)/(x**2*(b + c*x**2))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{6}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^6/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^6/(c*x^4 + b*x^2)^(3/2), x)