3.996 \(\int (d x)^m \left (c x^2\right )^p (a+b x)^{-2-m-2 p} \, dx\)

Optimal. Leaf size=39 \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]

[Out]

(x*(d*x)^m*(c*x^2)^p*(a + b*x)^(-1 - m - 2*p))/(a*(1 + m + 2*p))

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Rubi [A]  time = 0.0326008, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d*x)^m*(c*x^2)^p*(a + b*x)^(-1 - m - 2*p))/(a*(1 + m + 2*p))

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Rubi in Sympy [A]  time = 18.3341, size = 46, normalized size = 1.18 \[ \frac{\left (c x^{2}\right )^{p} \left (d x\right )^{- 2 p} \left (d x\right )^{m + 2 p + 1} \left (a + b x\right )^{- m - 2 p - 1}}{a d \left (m + 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x**2)**p*(d*x)**(-2*p)*(d*x)**(m + 2*p + 1)*(a + b*x)**(-m - 2*p - 1)/(a*d*(m
+ 2*p + 1))

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Mathematica [A]  time = 0.0527546, size = 39, normalized size = 1. \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a m+2 a p+a} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d*x)^m*(c*x^2)^p*(a + b*x)^(-1 - m - 2*p))/(a + a*m + 2*a*p)

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Maple [A]  time = 0.004, size = 40, normalized size = 1. \[{\frac{x \left ( dx \right ) ^{m} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{-1-m-2\,p}}{a \left ( 1+m+2\,p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(d*x)^m*(c*x^2)^p*(b*x+a)^(-1-m-2*p)/a/(1+m+2*p)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2)^p*(b*x + a)^(-m - 2*p - 2)*(d*x)^m, x)

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Fricas [A]  time = 0.235686, size = 77, normalized size = 1.97 \[ \frac{{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{-m - 2 \, p - 2} \left (d x\right )^{m} e^{\left (2 \, p \log \left (d x\right ) + p \log \left (\frac{c}{d^{2}}\right )\right )}}{a m + 2 \, a p + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x^2 + a*x)*(b*x + a)^(-m - 2*p - 2)*(d*x)^m*e^(2*p*log(d*x) + p*log(c/d^2))/(
a*m + 2*a*p + a)

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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((d*x)**m*(c*x**2)**p*(b*x+a)**(-2-m-2*p),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2)^p*(b*x + a)^(-m - 2*p - 2)*(d*x)^m, x)