3.714 \(\int \frac{(d+e x)^m}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=304 \[ -\frac{(d+e x)^{m+1} \left (\sqrt{-a} \sqrt{c} d e m+a e^2 (1-m)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 (-a)^{3/2} (m+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{m+1} \left (-\sqrt{-a} \sqrt{c} d e m+a e^2 (1-m)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 (-a)^{3/2} (m+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{m+1} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.916433, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{(d+e x)^{m+1} \left (\sqrt{-a} \sqrt{c} d e m+a e^2 (1-m)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 (-a)^{3/2} (m+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{m+1} \left (-\sqrt{-a} \sqrt{c} d e m+a e^2 (1-m)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 (-a)^{3/2} (m+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{m+1} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m/(a + c*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 95.0137, size = 246, normalized size = 0.81 \[ \frac{\left (d + e x\right )^{m + 1} \left (a e + c d x\right )}{2 a \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a \sqrt{c} d e m + \sqrt{- a} \left (a e^{2} \left (- m + 1\right ) + c d^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{4 a^{2} \left (m + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{m + 1} \left (- a \sqrt{c} d e m + \sqrt{- a} \left (a e^{2} \left (- m + 1\right ) + c d^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{4 a^{2} \left (m + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m/(c*x**2+a)**2,x)

[Out]

(d + e*x)**(m + 1)*(a*e + c*d*x)/(2*a*(a + c*x**2)*(a*e**2 + c*d**2)) + (d + e*x
)**(m + 1)*(a*sqrt(c)*d*e*m + sqrt(-a)*(a*e**2*(-m + 1) + c*d**2))*hyper((1, m +
1), (m + 2,), sqrt(c)*(d + e*x)/(sqrt(c)*d + e*sqrt(-a)))/(4*a**2*(m + 1)*(a*e*
*2 + c*d**2)*(sqrt(c)*d + e*sqrt(-a))) - (d + e*x)**(m + 1)*(-a*sqrt(c)*d*e*m +
sqrt(-a)*(a*e**2*(-m + 1) + c*d**2))*hyper((1, m + 1), (m + 2,), sqrt(c)*(d + e*
x)/(sqrt(c)*d - e*sqrt(-a)))/(4*a**2*(m + 1)*(a*e**2 + c*d**2)*(sqrt(c)*d - e*sq
rt(-a)))

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Mathematica [A]  time = 0.0712836, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (a+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^m/(a + c*x^2)^2,x]

[Out]

Integrate[(d + e*x)^m/(a + c*x^2)^2, x]

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(c*x^2+a)^2,x)

[Out]

int((e*x+d)^m/(c*x^2+a)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(c*x^2 + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^m/(c^2*x^4 + 2*a*c*x^2 + a^2), 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+d)**m/(c*x**2+a)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(c*x^2 + a)^2, x)