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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]