3.1491 \(\int \frac{(A+B x) (d+e x)^m}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=361 \[ \frac{(d+e x)^{m+1} \left (a e m (a B e+A c d)-\sqrt{-a} \sqrt{c} \left (A \left (a e^2 (1-m)+c d^2\right )+a B d e m\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 a^2 \sqrt{c} (m+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{m+1} \left (\sqrt{-a} \sqrt{c} \left (A \left (a e^2 (1-m)+c d^2\right )+a B d e m\right )+a e m (a B e+A c d)\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 a^2 \sqrt{c} (m+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{(d+e x)^{m+1} (a (B d-A e)-x (a B e+A c d))}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

[Out]

-((d + e*x)^(1 + m)*(a*(B*d - A*e) - (A*c*d + a*B*e)*x))/(2*a*(c*d^2 + a*e^2)*(a
+ c*x^2)) + ((a*e*(A*c*d + a*B*e)*m - Sqrt[-a]*Sqrt[c]*(A*(c*d^2 + a*e^2*(1 - m
)) + a*B*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^2*Sqrt[c]*(Sqrt[c]*d - Sqrt[-a]*e)*(c*
d^2 + a*e^2)*(1 + m)) + ((a*e*(A*c*d + a*B*e)*m + Sqrt[-a]*Sqrt[c]*(A*(c*d^2 + a
*e^2*(1 - m)) + a*B*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^2*Sqrt[c]*(Sqrt[c]*d + Sqrt
[-a]*e)*(c*d^2 + a*e^2)*(1 + m))

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Rubi [A]  time = 1.21069, antiderivative size = 359, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(d+e x)^{m+1} \left (a e m (a B e+A c d)-\sqrt{-a} \sqrt{c} \left (a A e^2 (1-m)+a B d e m+A c d^2\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{4 a^2 \sqrt{c} (m+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )}+\frac{(d+e x)^{m+1} \left (\sqrt{-a} \sqrt{c} \left (a A e^2 (1-m)+a B d e m+A c d^2\right )+a e m (a B e+A c d)\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{4 a^2 \sqrt{c} (m+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{(d+e x)^{m+1} (a (B d-A e)-x (a B e+A c d))}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)^(1 + m)*(a*(B*d - A*e) - (A*c*d + a*B*e)*x))/(2*a*(c*d^2 + a*e^2)*(a
+ c*x^2)) + ((a*e*(A*c*d + a*B*e)*m - Sqrt[-a]*Sqrt[c]*(A*c*d^2 + a*A*e^2*(1 -
m) + a*B*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^2*Sqrt[c]*(Sqrt[c]*d - Sqrt[-a]*e)*(c*
d^2 + a*e^2)*(1 + m)) + ((a*e*(A*c*d + a*B*e)*m + Sqrt[-a]*Sqrt[c]*(A*c*d^2 + a*
A*e^2*(1 - m) + a*B*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^2*Sqrt[c]*(Sqrt[c]*d + Sqrt
[-a]*e)*(c*d^2 + a*e^2)*(1 + m))

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Rubi in Sympy [A]  time = 123.19, size = 306, normalized size = 0.85 \[ \frac{\left (d + e x\right )^{m + 1} \left (a \left (A e - B d\right ) + x \left (A c d + B a e\right )\right )}{2 a \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e m \left (A c d + B a e\right ) + \sqrt{c} \sqrt{- a} \left (A \left (a e^{2} \left (- m + 1\right ) + c d^{2}\right ) + B a d e m\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{4 a^{2} \sqrt{c} \left (m + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{m + 1} \left (- a e m \left (A c d + B a e\right ) + \sqrt{c} \sqrt{- a} \left (A \left (a e^{2} \left (- m + 1\right ) + c d^{2}\right ) + B a d e m\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{4 a^{2} \sqrt{c} \left (m + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.162492, size = 0, normalized size = 0. \[ \int \frac{(A+B x) (d+e x)^m}{\left (a+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)*(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((B*x+A)*(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. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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