3.443 \(\int \frac{(d+e x)^m}{b x+c x^2} \, dx\)

Optimal. Leaf size=93 \[ \frac{c (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{b (m+1) (c d-b e)}-\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{b d (m+1)} \]

[Out]

(c*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e
)])/(b*(c*d - b*e)*(1 + m)) - ((d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 +
m, 1 + (e*x)/d])/(b*d*(1 + m))

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Rubi [A]  time = 0.12379, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{b (m+1) (c d-b e)}-\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{b d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e
)])/(b*(c*d - b*e)*(1 + m)) - ((d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 +
m, 1 + (e*x)/d])/(b*d*(1 + m))

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Rubi in Sympy [A]  time = 15.6945, size = 70, normalized size = 0.75 \[ - \frac{c \left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (- d - e x\right )}{b e - c d}} \right )}}{b \left (m + 1\right ) \left (b e - c d\right )} - \frac{\left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{1 + \frac{e x}{d}} \right )}}{b d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*(d + e*x)**(m + 1)*hyper((1, m + 1), (m + 2,), c*(-d - e*x)/(b*e - c*d))/(b*(
m + 1)*(b*e - c*d)) - (d + e*x)**(m + 1)*hyper((1, m + 1), (m + 2,), 1 + e*x/d)/
(b*d*(m + 1))

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Mathematica [A]  time = 0.116603, size = 98, normalized size = 1.05 \[ \frac{(d+e x)^m \left (\frac{c (d+e x) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{(m+1) (c d-b e)}+\frac{\left (\frac{d}{e x}+1\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{d}{e x}\right )}{m}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^m*((c*(d + e*x)*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d
- b*e)])/((c*d - b*e)*(1 + m)) + Hypergeometric2F1[-m, -m, 1 - m, -(d/(e*x))]/(
m*(1 + d/(e*x))^m)))/b

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{c{x}^{2}+bx}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(c*x^2+b*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m/(c*x**2+b*x),x)

[Out]

Integral((d + e*x)**m/(x*(b + c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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