3.1729 \(\int \frac{(d+e x)^m}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{(a+b x) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0983065, antiderivative size = 76, 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 \[ -\frac{(a+b x) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.1337, size = 63, normalized size = 0.83 \[ \frac{\left (d + e x\right )^{m + 1} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{\left (a + b x\right ) \left (m + 1\right ) \left (a e - b d\right )} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0649249, size = 67, normalized size = 0.88 \[ -\frac{(a+b x) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) \sqrt{(a+b x)^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,abx+{a}^{2}}}}}\, dx \]

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)^(1/2),x)

[Out]

int((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

Integral((d + e*x)**m/sqrt((a + b*x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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