3.393 \(\int \frac{x^7 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=312 \[ \frac{\left (\frac{b \left (b^2-3 a c\right )}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{2 c (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{\left (-\frac{b \left (b^2-3 a c\right )}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{(b e+c d) \left (d+e x^2\right )^{q+1}}{2 c^2 e^2 (q+1)}+\frac{\left (d+e x^2\right )^{q+2}}{2 c e^2 (q+2)} \]

[Out]

-((c*d + b*e)*(d + e*x^2)^(1 + q))/(2*c^2*e^2*(1 + q)) + (d + e*x^2)^(2 + q)/(2*
c*e^2*(2 + q)) + ((a - b^2/c + (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*(d + e*x
^2)^(1 + q)*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - b*e +
Sqrt[b^2 - 4*a*c]*e)])/(2*c*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(1 + q)) + ((a -
b^2/c - (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*(d + e*x^2)^(1 + q)*Hypergeome
tric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])
/(2*c*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(1 + q))

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Rubi [A]  time = 1.80059, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\left (\frac{b \left (b^2-3 a c\right )}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{2 c (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{\left (-\frac{b \left (b^2-3 a c\right )}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{(b e+c d) \left (d+e x^2\right )^{q+1}}{2 c^2 e^2 (q+1)}+\frac{\left (d+e x^2\right )^{q+2}}{2 c e^2 (q+2)} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]

[Out]

-((c*d + b*e)*(d + e*x^2)^(1 + q))/(2*c^2*e^2*(1 + q)) + (d + e*x^2)^(2 + q)/(2*
c*e^2*(2 + q)) + ((a - b^2/c + (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*(d + e*x
^2)^(1 + q)*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - b*e +
Sqrt[b^2 - 4*a*c]*e)])/(2*c*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(1 + q)) + ((a -
b^2/c - (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*(d + e*x^2)^(1 + q)*Hypergeome
tric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^2))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])
/(2*c*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(1 + q))

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Rubi in Sympy [A]  time = 96.978, size = 289, normalized size = 0.93 \[ \frac{\left (d + e x^{2}\right )^{q + 2}}{2 c e^{2} \left (q + 2\right )} + \frac{\left (d + e x^{2}\right )^{q + 1} \left (b \left (- 3 a c + b^{2}\right ) - \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x^{2}\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{2} \left (q + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c d - e \left (b - \sqrt{- 4 a c + b^{2}}\right )\right )} - \frac{\left (d + e x^{2}\right )^{q + 1} \left (b \left (- 3 a c + b^{2}\right ) + \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x^{2}\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{2} \left (q + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )\right )} - \frac{\left (d + e x^{2}\right )^{q + 1} \left (b e + c d\right )}{2 c^{2} e^{2} \left (q + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(e*x**2+d)**q/(c*x**4+b*x**2+a),x)

[Out]

(d + e*x**2)**(q + 2)/(2*c*e**2*(q + 2)) + (d + e*x**2)**(q + 1)*(b*(-3*a*c + b*
*2) - sqrt(-4*a*c + b**2)*(-a*c + b**2))*hyper((1, q + 1), (q + 2,), c*(-2*d - 2
*e*x**2)/(b*e - 2*c*d - e*sqrt(-4*a*c + b**2)))/(2*c**2*(q + 1)*sqrt(-4*a*c + b*
*2)*(2*c*d - e*(b - sqrt(-4*a*c + b**2)))) - (d + e*x**2)**(q + 1)*(b*(-3*a*c +
b**2) + sqrt(-4*a*c + b**2)*(-a*c + b**2))*hyper((1, q + 1), (q + 2,), c*(-2*d -
2*e*x**2)/(b*e - 2*c*d + e*sqrt(-4*a*c + b**2)))/(2*c**2*(q + 1)*sqrt(-4*a*c +
b**2)*(2*c*d - e*(b + sqrt(-4*a*c + b**2)))) - (d + e*x**2)**(q + 1)*(b*e + c*d)
/(2*c**2*e**2*(q + 1))

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Mathematica [A]  time = 1.35108, size = 367, normalized size = 1.18 \[ \frac{2^{-q-2} \left (d+e x^2\right )^q \left (\left (b^2 \sqrt{e^2 \left (b^2-4 a c\right )}-a c \sqrt{e^2 \left (b^2-4 a c\right )}+3 a b c e+b^3 (-e)\right ) \left (\frac{c \left (d+e x^2\right )}{-\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x^2}\right )^{-q} \, _2F_1\left (-q,-q;1-q;\frac{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{-2 c e x^2-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+\left (b^2 \sqrt{e^2 \left (b^2-4 a c\right )}-a c \sqrt{e^2 \left (b^2-4 a c\right )}-3 a b c e+b^3 e\right ) \left (\frac{c \left (d+e x^2\right )}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x^2}\right )^{-q} \, _2F_1\left (-q,-q;1-q;\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c e x^2+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{c^3 q \sqrt{e^2 \left (b^2-4 a c\right )}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^7*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]

[Out]

(2^(-2 - q)*(d + e*x^2)^q*(((-(b^3*e) + 3*a*b*c*e + b^2*Sqrt[(b^2 - 4*a*c)*e^2]
- a*c*Sqrt[(b^2 - 4*a*c)*e^2])*Hypergeometric2F1[-q, -q, 1 - q, (2*c*d - b*e + S
qrt[(b^2 - 4*a*c)*e^2])/(-(b*e) + Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*e*x^2)])/((c*(d
+ e*x^2))/(b*e - Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*e*x^2))^q + ((b^3*e - 3*a*b*c*e +
b^2*Sqrt[(b^2 - 4*a*c)*e^2] - a*c*Sqrt[(b^2 - 4*a*c)*e^2])*Hypergeometric2F1[-q
, -q, 1 - q, (-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(b*e + Sqrt[(b^2 - 4*a*c)*
e^2] + 2*c*e*x^2)])/((c*(d + e*x^2))/(b*e + Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*e*x^2)
)^q))/(c^3*Sqrt[(b^2 - 4*a*c)*e^2]*q)

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{\frac{{x}^{7} \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(e*x^2+d)^q/(c*x^4+b*x^2+a),x)

[Out]

int(x^7*(e*x^2+d)^q/(c*x^4+b*x^2+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{q} x^{7}}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q*x^7/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^q*x^7/(c*x^4 + b*x^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x^{2} + d\right )}^{q} x^{7}}{c x^{4} + b x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q*x^7/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)^q*x^7/(c*x^4 + b*x^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(e*x**2+d)**q/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{q} x^{7}}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q*x^7/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^q*x^7/(c*x^4 + b*x^2 + a), x)