3.2076 \(\int \frac{(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0722288, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.4865, size = 41, normalized size = 0.76 \[ \frac{\left (d + e x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, m \\ m + 1 \end{matrix}\middle |{\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}} \right )}}{m \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.120921, size = 87, normalized size = 1.61 \[ -\frac{(d+e x)^m \left (c d m (d+e x) \, _2F_1\left (1,m+1;m+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )+(m+1) \left (c d^2-a e^2\right )\right )}{m (m+1) \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.164, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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