3.1045 \(\int x^6 \left (a+b x^2\right )^p \, dx\)

Optimal. Leaf size=40 \[ \frac{x^7 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+\frac{9}{2};\frac{9}{2};-\frac{b x^2}{a}\right )}{7 a} \]

[Out]

(x^7*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 9/2 + p, 9/2, -((b*x^2)/a)])/(7*a)

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Rubi [A]  time = 0.0413751, antiderivative size = 49, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{7} x^7 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x^2)^p,x]

[Out]

(x^7*(a + b*x^2)^p*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(7*(1 + (b*x^2
)/a)^p)

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Rubi in Sympy [A]  time = 7.94813, size = 37, normalized size = 0.92 \[ \frac{x^{7} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**7*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*hyper((-p, 7/2), (9/2,), -b*x**2/a)/7

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Mathematica [A]  time = 0.0290004, size = 49, normalized size = 1.22 \[ \frac{1}{7} x^7 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^6*(a + b*x^2)^p,x]

[Out]

(x^7*(a + b*x^2)^p*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(7*(1 + (b*x^2
)/a)^p)

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{x}^{6} \left ( b{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x^2+a)^p,x)

[Out]

int(x^6*(b*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p} x^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^p*x^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^p*x^6, x)

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Sympy [A]  time = 59.0101, size = 26, normalized size = 0.65 \[ \frac{a^{p} x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{2}, - p \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(b*x**2+a)**p,x)

[Out]

a**p*x**7*hyper((7/2, -p), (9/2,), b*x**2*exp_polar(I*pi)/a)/7

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^p*x^6, x)