3.1417 \(\int \frac{(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=70 \[ -\frac{32 c^2 (d (b+2 c x))^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^3} \]

[Out]

(-32*c^2*(d*(b + 2*c*x))^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, (b +
2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)^3*d*(1 + m))

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Rubi [A]  time = 0.129752, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{32 c^2 (d (b+2 c x))^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(-32*c^2*(d*(b + 2*c*x))^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, (b +
2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)^3*d*(1 + m))

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Rubi in Sympy [A]  time = 26.2462, size = 65, normalized size = 0.93 \[ - \frac{32 c^{2} \left (b d + 2 c d x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{d \left (m + 1\right ) \left (- 4 a c + b^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*c**2*(b*d + 2*c*d*x)**(m + 1)*hyper((3, m/2 + 1/2), (m/2 + 3/2,), -(b + 2*c*
x)**2/(4*a*c - b**2))/(d*(m + 1)*(-4*a*c + b**2)**3)

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Mathematica [A]  time = 0.0944244, size = 0, normalized size = 0. \[ \int \frac{(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.234, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2\,cdx+bd \right ) ^{m}}{ \left ( c{x}^{2}+bx+a \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((2*c*d*x+b*d)^m/(c*x^2+b*x+a)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (2 \, c d x + b d\right )}^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((2*c*d*x + b*d)^m/(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^
2*b*x + (b^3 + 6*a*b*c)*x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^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((2*c*d*x+b*d)**m/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a)^3, x)