3.398 \(\int \frac{\left (d+e x^2\right )^q}{x^3 \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=321 \[ -\frac{c \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{2 a^2 (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 a^2 (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}+\frac{b \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{e x^2}{d}+1\right )}{2 a^2 d (q+1)}+\frac{e \left (d+e x^2\right )^{q+1} \, _2F_1\left (2,q+1;q+2;\frac{e x^2}{d}+1\right )}{2 a d^2 (q+1)} \]

[Out]

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

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Rubi [A]  time = 1.57045, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{c \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{2 a^2 (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{c \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 a^2 (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}+\frac{b \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{e x^2}{d}+1\right )}{2 a^2 d (q+1)}+\frac{e \left (d+e x^2\right )^{q+1} \, _2F_1\left (2,q+1;q+2;\frac{e x^2}{d}+1\right )}{2 a d^2 (q+1)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)^q/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 141.294, size = 303, normalized size = 0.94 \[ \frac{e \left (d + e x^{2}\right )^{q + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, q + 1 \\ q + 2 \end{matrix}\middle |{1 + \frac{e x^{2}}{d}} \right )}}{2 a d^{2} \left (q + 1\right )} + \frac{b \left (d + e x^{2}\right )^{q + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{1 + \frac{e x^{2}}{d}} \right )}}{2 a^{2} d \left (q + 1\right )} - \frac{c \left (d + e x^{2}\right )^{q + 1} \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x^{2}\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a^{2} \left (q + 1\right ) \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )} - \frac{c \left (d + e x^{2}\right )^{q + 1} \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x^{2}\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a^{2} \left (q + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c d - e \left (b - \sqrt{- 4 a c + b^{2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)**q/x**3/(c*x**4+b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.988913, size = 420, normalized size = 1.31 \[ \frac{2^{-q-2} \left (d+e x^2\right )^q \left (\left (b \sqrt{e^2 \left (b^2-4 a c\right )}-2 a c e+b^2 e\right ) \left (\frac{c \left (d+e x^2\right )}{-\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x^2}\right )^{-q} \, _2F_1\left (-q,-q;1-q;\frac{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{-2 c e x^2-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+\left (b \sqrt{e^2 \left (b^2-4 a c\right )}+2 a c e+b^2 (-e)\right ) \left (\frac{c \left (d+e x^2\right )}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x^2}\right )^{-q} \, _2F_1\left (-q,-q;1-q;\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c e x^2+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{a^2 q \sqrt{e^2 \left (b^2-4 a c\right )}}+\frac{\left (d+e x^2\right )^q \left (\frac{d}{e x^2}+1\right )^{-q} \left (a q \, _2F_1\left (1-q,-q;2-q;-\frac{d}{e x^2}\right )-b (q-1) x^2 \, _2F_1\left (-q,-q;1-q;-\frac{d}{e x^2}\right )\right )}{2 a^2 (q-1) q x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2)^q/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Maple [F]  time = 0.121, size = 0, normalized size = 0. \[ \int{\frac{ \left ( e{x}^{2}+d \right ) ^{q}}{{x}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^q/x^3/(c*x^4+b*x^2+a),x)

[Out]

int((e*x^2+d)^q/x^3/(c*x^4+b*x^2+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{q}}{{\left (c x^{4} + b x^{2} + a\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q/((c*x^4 + b*x^2 + a)*x^3),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^q/((c*x^4 + b*x^2 + a)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x^{2} + d\right )}^{q}}{c x^{7} + b x^{5} + a x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q/((c*x^4 + b*x^2 + a)*x^3),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)^q/(c*x^7 + b*x^5 + a*x^3), 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((e*x**2+d)**q/x**3/(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. \[ \int \frac{{\left (e x^{2} + d\right )}^{q}}{{\left (c x^{4} + b x^{2} + a\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q/((c*x^4 + b*x^2 + a)*x^3),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^q/((c*x^4 + b*x^2 + a)*x^3), x)