3.1492 \(\int \frac{(A+B x) (d+e x)^{1+m}}{a+c x^2} \, dx\)

Optimal. Leaf size=202 \[ -\frac{\left (\sqrt{-a} A \sqrt{c}+a B\right ) (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a \sqrt{c} (m+2) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{\left (\frac{\sqrt{-a} B}{\sqrt{c}}+A\right ) (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{-a} (m+2) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

[Out]

-((a*B + Sqrt[-a]*A*Sqrt[c])*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m
, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(2*a*Sqrt[c]*(Sqrt[c]*d - Sqrt[
-a]*e)*(2 + m)) - ((A + (Sqrt[-a]*B)/Sqrt[c])*(d + e*x)^(2 + m)*Hypergeometric2F
1[1, 2 + m, 3 + m, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*Sqrt[-a]*(S
qrt[c]*d + Sqrt[-a]*e)*(2 + m))

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Rubi [A]  time = 0.401852, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\left (\sqrt{-a} A \sqrt{c}+a B\right ) (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a \sqrt{c} (m+2) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{\left (\frac{\sqrt{-a} B}{\sqrt{c}}+A\right ) (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{-a} (m+2) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(1 + m))/(a + c*x^2),x]

[Out]

-((a*B + Sqrt[-a]*A*Sqrt[c])*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m
, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(2*a*Sqrt[c]*(Sqrt[c]*d - Sqrt[
-a]*e)*(2 + m)) - ((A + (Sqrt[-a]*B)/Sqrt[c])*(d + e*x)^(2 + m)*Hypergeometric2F
1[1, 2 + m, 3 + m, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(2*Sqrt[-a]*(S
qrt[c]*d + Sqrt[-a]*e)*(2 + m))

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Rubi in Sympy [A]  time = 48.8955, size = 167, normalized size = 0.83 \[ - \frac{\left (d + e x\right )^{m + 2} \left (- A \sqrt{c} \sqrt{- a} + B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 a \sqrt{c} \left (m + 2\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{m + 2} \left (A \sqrt{c} \sqrt{- a} + B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 a \sqrt{c} \left (m + 2\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(1+m)/(c*x**2+a),x)

[Out]

-(d + e*x)**(m + 2)*(-A*sqrt(c)*sqrt(-a) + B*a)*hyper((1, m + 2), (m + 3,), sqrt
(c)*(d + e*x)/(sqrt(c)*d + e*sqrt(-a)))/(2*a*sqrt(c)*(m + 2)*(sqrt(c)*d + e*sqrt
(-a))) - (d + e*x)**(m + 2)*(A*sqrt(c)*sqrt(-a) + B*a)*hyper((1, m + 2), (m + 3,
), sqrt(c)*(d + e*x)/(sqrt(c)*d - e*sqrt(-a)))/(2*a*sqrt(c)*(m + 2)*(sqrt(c)*d -
e*sqrt(-a)))

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Mathematica [C]  time = 0.892184, size = 303, normalized size = 1.5 \[ \frac{(d+e x)^m \left (\left (A \sqrt{c}+i \sqrt{a} B\right ) \left (\sqrt{a} e-i \sqrt{c} d\right ) \left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x-i \sqrt{a}\right )}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{\sqrt{c} d+i \sqrt{a} e}{i \sqrt{a} e-\sqrt{c} e x}\right )+\left (A \sqrt{c}-i \sqrt{a} B\right ) \left (\sqrt{a} e+i \sqrt{c} d\right ) \left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} x e+i \sqrt{a} e}\right )+\frac{2 \sqrt{a} B \sqrt{c} m (d+e x)}{m+1}\right )}{2 \sqrt{a} c^{3/2} m} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(1 + m))/(a + c*x^2),x]

[Out]

((d + e*x)^m*((2*Sqrt[a]*B*Sqrt[c]*m*(d + e*x))/(1 + m) + ((I*Sqrt[a]*B + A*Sqrt
[c])*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-m, -m, 1 - m, (Sqrt[c]*d +
I*Sqrt[a]*e)/(I*Sqrt[a]*e - Sqrt[c]*e*x)])/((Sqrt[c]*(d + e*x))/(e*((-I)*Sqrt[a]
+ Sqrt[c]*x)))^m + (((-I)*Sqrt[a]*B + A*Sqrt[c])*(I*Sqrt[c]*d + Sqrt[a]*e)*Hype
rgeometric2F1[-m, -m, 1 - m, -((Sqrt[c]*d - I*Sqrt[a]*e)/(I*Sqrt[a]*e + Sqrt[c]*
e*x))])/((Sqrt[c]*(d + e*x))/(e*(I*Sqrt[a] + Sqrt[c]*x)))^m))/(2*Sqrt[a]*c^(3/2)
*m)

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{1+m}}{c{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(1+m)/(c*x^2+a),x)

[Out]

int((B*x+A)*(e*x+d)^(1+m)/(c*x^2+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m + 1}}{c x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(m + 1)/(c*x^2 + a),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(m + 1)/(c*x^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m + 1}}{c x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(m + 1)/(c*x^2 + a),x, algorithm="fricas")

[Out]

integral((B*x + A)*(e*x + d)^(m + 1)/(c*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((B*x+A)*(e*x+d)**(1+m)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m + 1}}{c x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(m + 1)/(c*x^2 + a),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^(m + 1)/(c*x^2 + a), x)