Optimal. Leaf size=79 \[ -\frac{e^2 (a+b x) (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
[Out]
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Rubi [A] time = 0.129179, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{e^2 (a+b x) (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 19.4327, size = 68, normalized size = 0.86 \[ \frac{e^{2} \left (d + e x\right )^{m + 1} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{\left (a + b x\right ) \left (m + 1\right ) \left (a e - b d\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0690239, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [F] time = 0.108, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{m}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**m/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")
[Out]