3.1490 \(\int \frac{(A+B x) (d+e x)^m}{a+c x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.471976, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\left (\sqrt{-a} A \sqrt{c}+a B\right ) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 a \sqrt{c} (m+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{\left (\frac{\sqrt{-a} B}{\sqrt{c}}+A\right ) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{-a} (m+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.7332, size = 167, normalized size = 0.83 \[ - \frac{\left (d + e x\right )^{m + 1} \left (- A \sqrt{c} \sqrt{- a} + B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 a \sqrt{c} \left (m + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{m + 1} \left (A \sqrt{c} \sqrt{- a} + B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 a \sqrt{c} \left (m + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.426793, size = 241, normalized size = 1.19 \[ \frac{(d+e x)^m \left (\left (\sqrt{a} B-i A \sqrt{c}\right ) \left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x-i \sqrt{a}\right )}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{\sqrt{c} d+i \sqrt{a} e}{i \sqrt{a} e-\sqrt{c} e x}\right )+\left (\sqrt{a} B+i A \sqrt{c}\right ) \left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} x e+i \sqrt{a} e}\right )\right )}{2 \sqrt{a} c m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{c{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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