3.1892 \(\int \frac{(A+B x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=169 \[ \frac{e (a+b x) (d+e x)^{m+1} (b (2 B d-A e (1-m))-a B e (m+1)) \, _2F_1\left (2,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{2 b (m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{(A b-a B) (d+e x)^{m+1}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

-((A*b - a*B)*(d + e*x)^(1 + m))/(2*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (e*(b*(2*B*d - A*e*(1 - m)) - a*B*e*(1 + m))*(a + b*x)*(d + e*x)^(1
+ m)*Hypergeometric2F1[2, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a*e)])/(2*b*(b*d -
a*e)^3*(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.396212, antiderivative size = 168, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{e (a+b x) (d+e x)^{m+1} (-a B e (m+1)-A b e (1-m)+2 b B d) \, _2F_1\left (2,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{2 b (m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{(A b-a B) (d+e x)^{m+1}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

-((A*b - a*B)*(d + e*x)^(1 + m))/(2*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (e*(2*b*B*d - A*b*e*(1 - m) - a*B*e*(1 + m))*(a + b*x)*(d + e*x)^(1
+ m)*Hypergeometric2F1[2, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a*e)])/(2*b*(b*d -
a*e)^3*(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 37.8569, size = 143, normalized size = 0.85 \[ - \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{m + 1}}{2 b e \left (- m + 1\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{e \left (d + e x\right )^{m + 1} \left (- A b e \left (- m + 1\right ) + B \left (- a e \left (m + 1\right ) + 2 b d\right )\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{b \left (a + b x\right ) \left (- m + 1\right ) \left (m + 1\right ) \left (a e - b d\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**m/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

-B*(2*a + 2*b*x)*(d + e*x)**(m + 1)/(2*b*e*(-m + 1)*(a**2 + 2*a*b*x + b**2*x**2)
**(3/2)) - e*(d + e*x)**(m + 1)*(-A*b*e*(-m + 1) + B*(-a*e*(m + 1) + 2*b*d))*sqr
t(a**2 + 2*a*b*x + b**2*x**2)*hyper((3, m + 1), (m + 2,), b*(-d - e*x)/(a*e - b*
d))/(b*(a + b*x)*(-m + 1)*(m + 1)*(a*e - b*d)**3)

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Mathematica [A]  time = 0.203384, size = 0, normalized size = 0. \[ \int \frac{(A+B x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[((A + B*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

Integrate[((A + B*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2), x]

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \[ \int{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

int((B*x+A)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

integral((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**m/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^(3/2), x)