3.10.89 \(\int x (c x^2)^p (a+b x)^{-3-2 p} \, dx\) [989]

Optimal. Leaf size=33 \[ \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (1+p)}}{2 a (1+p)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 37} \begin {gather*} \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (p+1)}}{2 a (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(c*x^2)^p*(a + b*x)^(-3 - 2*p),x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int x \left (c x^2\right )^p (a+b x)^{-3-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{1+2 p} (a+b x)^{-3-2 p} \, dx\\ &=\frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (1+p)}}{2 a (1+p)}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 32, normalized size = 0.97 \begin {gather*} \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2-2 p}}{a (2+2 p)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(c*x^2)^p*(a + b*x)^(-3 - 2*p),x]

[Out]

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

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Maple [A]
time = 0.14, size = 32, normalized size = 0.97

method	result	size
gosper	\(\frac {x^{2} \left (b x +a \right )^{-2-2 p} \left (c \,x^{2}\right )^{p}}{2 a \left (1+p \right )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^2)^p*(b*x+a)^(-3-2*p),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2)^p*(b*x+a)^(-3-2*p),x, algorithm="maxima")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-2*p - 3)*x, x)

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Fricas [A]
time = 0.87, size = 38, normalized size = 1.15 \begin {gather*} \frac {{\left (b x^{3} + a x^{2}\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 3}}{2 \, {\left (a p + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2)^p*(b*x+a)^(-3-2*p),x, algorithm="fricas")

[Out]

1/2*(b*x^3 + a*x^2)*(c*x^2)^p*(b*x + a)^(-2*p - 3)/(a*p + a)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (27) = 54\).
time = 145.45, size = 1409, normalized size = 42.70 \begin {gather*} \begin {cases} - \frac {\left (b x\right )^{- 2 p} \left (c x^{2}\right )^{p}}{b^{3} x} & \text {for}\: a = 0 \\\frac {0^{- 2 p - 3} x^{2} \left (c x^{2}\right )^{p}}{2 p + 2} & \text {for}\: a = - b x \\\frac {x^{2} \left (c x^{2}\right )^{p} \left (0^{\frac {1}{p}}\right )^{- 2 p - 3}}{2 p + 2} & \text {for}\: a = 0^{\frac {1}{p}} - b x \\\frac {\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x \right )}}{a}}{c} & \text {for}\: p = -1 \\\frac {a^{4} x^{2} \left (c x^{2}\right )^{p}}{2 a^{7} p \left (a + b x\right )^{2 p} + 2 a^{7} \left (a + b x\right )^{2 p} + 12 a^{6} b p x \left (a + b x\right )^{2 p} + 12 a^{6} b x \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} x^{2} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} x^{3} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} p x^{4} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} x^{4} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} p x^{5} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} x^{5} \left (a + b x\right )^{2 p} + 2 a b^{6} p x^{6} \left (a + b x\right )^{2 p} + 2 a b^{6} x^{6} \left (a + b x\right )^{2 p}} + \frac {3 a^{3} b x^{3} \left (c x^{2}\right )^{p}}{2 a^{7} p \left (a + b x\right )^{2 p} + 2 a^{7} \left (a + b x\right )^{2 p} + 12 a^{6} b p x \left (a + b x\right )^{2 p} + 12 a^{6} b x \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} x^{2} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} x^{3} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} p x^{4} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} x^{4} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} p x^{5} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} x^{5} \left (a + b x\right )^{2 p} + 2 a b^{6} p x^{6} \left (a + b x\right )^{2 p} + 2 a b^{6} x^{6} \left (a + b x\right )^{2 p}} + \frac {3 a^{2} b^{2} x^{4} \left (c x^{2}\right )^{p}}{2 a^{7} p \left (a + b x\right )^{2 p} + 2 a^{7} \left (a + b x\right )^{2 p} + 12 a^{6} b p x \left (a + b x\right )^{2 p} + 12 a^{6} b x \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} x^{2} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} x^{3} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} p x^{4} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} x^{4} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} p x^{5} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} x^{5} \left (a + b x\right )^{2 p} + 2 a b^{6} p x^{6} \left (a + b x\right )^{2 p} + 2 a b^{6} x^{6} \left (a + b x\right )^{2 p}} + \frac {a b^{3} x^{5} \left (c x^{2}\right )^{p}}{2 a^{7} p \left (a + b x\right )^{2 p} + 2 a^{7} \left (a + b x\right )^{2 p} + 12 a^{6} b p x \left (a + b x\right )^{2 p} + 12 a^{6} b x \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} x^{2} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} x^{3} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} p x^{4} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} x^{4} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} p x^{5} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} x^{5} \left (a + b x\right )^{2 p} + 2 a b^{6} p x^{6} \left (a + b x\right )^{2 p} + 2 a b^{6} x^{6} \left (a + b x\right )^{2 p}} + \frac {b x^{3} \left (c x^{2}\right )^{p}}{2 a^{4} p \left (a + b x\right )^{2 p} + 2 a^{4} \left (a + b x\right )^{2 p} + 6 a^{3} b p x \left (a + b x\right )^{2 p} + 6 a^{3} b x \left (a + b x\right )^{2 p} + 6 a^{2} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 6 a^{2} b^{2} x^{2} \left (a + b x\right )^{2 p} + 2 a b^{3} p x^{3} \left (a + b x\right )^{2 p} + 2 a b^{3} x^{3} \left (a + b x\right )^{2 p}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x**2)**p*(b*x+a)**(-3-2*p),x)

[Out]

Piecewise((-(c*x**2)**p/(b**3*x*(b*x)**(2*p)), Eq(a, 0)), (0**(-2*p - 3)*x**2*(c*x**2)**p/(2*p + 2), Eq(a, -b*
x)), (x**2*(c*x**2)**p*(0**(1/p))**(-2*p - 3)/(2*p + 2), Eq(a, 0**(1/p) - b*x)), ((log(x)/a - log(a/b + x)/a)/
c, Eq(p, -1)), (a**4*x**2*(c*x**2)**p/(2*a**7*p*(a + b*x)**(2*p) + 2*a**7*(a + b*x)**(2*p) + 12*a**6*b*p*x*(a
+ b*x)**(2*p) + 12*a**6*b*x*(a + b*x)**(2*p) + 30*a**5*b**2*p*x**2*(a + b*x)**(2*p) + 30*a**5*b**2*x**2*(a + b
*x)**(2*p) + 40*a**4*b**3*p*x**3*(a + b*x)**(2*p) + 40*a**4*b**3*x**3*(a + b*x)**(2*p) + 30*a**3*b**4*p*x**4*(
a + b*x)**(2*p) + 30*a**3*b**4*x**4*(a + b*x)**(2*p) + 12*a**2*b**5*p*x**5*(a + b*x)**(2*p) + 12*a**2*b**5*x**
5*(a + b*x)**(2*p) + 2*a*b**6*p*x**6*(a + b*x)**(2*p) + 2*a*b**6*x**6*(a + b*x)**(2*p)) + 3*a**3*b*x**3*(c*x**
2)**p/(2*a**7*p*(a + b*x)**(2*p) + 2*a**7*(a + b*x)**(2*p) + 12*a**6*b*p*x*(a + b*x)**(2*p) + 12*a**6*b*x*(a +
b*x)**(2*p) + 30*a**5*b**2*p*x**2*(a + b*x)**(2*p) + 30*a**5*b**2*x**2*(a + b*x)**(2*p) + 40*a**4*b**3*p*x**3
*(a + b*x)**(2*p) + 40*a**4*b**3*x**3*(a + b*x)**(2*p) + 30*a**3*b**4*p*x**4*(a + b*x)**(2*p) + 30*a**3*b**4*x
**4*(a + b*x)**(2*p) + 12*a**2*b**5*p*x**5*(a + b*x)**(2*p) + 12*a**2*b**5*x**5*(a + b*x)**(2*p) + 2*a*b**6*p*
x**6*(a + b*x)**(2*p) + 2*a*b**6*x**6*(a + b*x)**(2*p)) + 3*a**2*b**2*x**4*(c*x**2)**p/(2*a**7*p*(a + b*x)**(2
*p) + 2*a**7*(a + b*x)**(2*p) + 12*a**6*b*p*x*(a + b*x)**(2*p) + 12*a**6*b*x*(a + b*x)**(2*p) + 30*a**5*b**2*p
*x**2*(a + b*x)**(2*p) + 30*a**5*b**2*x**2*(a + b*x)**(2*p) + 40*a**4*b**3*p*x**3*(a + b*x)**(2*p) + 40*a**4*b
**3*x**3*(a + b*x)**(2*p) + 30*a**3*b**4*p*x**4*(a + b*x)**(2*p) + 30*a**3*b**4*x**4*(a + b*x)**(2*p) + 12*a**
2*b**5*p*x**5*(a + b*x)**(2*p) + 12*a**2*b**5*x**5*(a + b*x)**(2*p) + 2*a*b**6*p*x**6*(a + b*x)**(2*p) + 2*a*b
**6*x**6*(a + b*x)**(2*p)) + a*b**3*x**5*(c*x**2)**p/(2*a**7*p*(a + b*x)**(2*p) + 2*a**7*(a + b*x)**(2*p) + 12
*a**6*b*p*x*(a + b*x)**(2*p) + 12*a**6*b*x*(a + b*x)**(2*p) + 30*a**5*b**2*p*x**2*(a + b*x)**(2*p) + 30*a**5*b
**2*x**2*(a + b*x)**(2*p) + 40*a**4*b**3*p*x**3*(a + b*x)**(2*p) + 40*a**4*b**3*x**3*(a + b*x)**(2*p) + 30*a**
3*b**4*p*x**4*(a + b*x)**(2*p) + 30*a**3*b**4*x**4*(a + b*x)**(2*p) + 12*a**2*b**5*p*x**5*(a + b*x)**(2*p) + 1
2*a**2*b**5*x**5*(a + b*x)**(2*p) + 2*a*b**6*p*x**6*(a + b*x)**(2*p) + 2*a*b**6*x**6*(a + b*x)**(2*p)) + b*x**
3*(c*x**2)**p/(2*a**4*p*(a + b*x)**(2*p) + 2*a**4*(a + b*x)**(2*p) + 6*a**3*b*p*x*(a + b*x)**(2*p) + 6*a**3*b*
x*(a + b*x)**(2*p) + 6*a**2*b**2*p*x**2*(a + b*x)**(2*p) + 6*a**2*b**2*x**2*(a + b*x)**(2*p) + 2*a*b**3*p*x**3
*(a + b*x)**(2*p) + 2*a*b**3*x**3*(a + b*x)**(2*p)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (33) = 66\).
time = 0.79, size = 72, normalized size = 2.18 \begin {gather*} \frac {\left (c x^{2}\right )^{p} b x^{3} e^{\left (-2 \, p \log \left (b x + a\right ) - 3 \, \log \left (b x + a\right )\right )} + \left (c x^{2}\right )^{p} a x^{2} e^{\left (-2 \, p \log \left (b x + a\right ) - 3 \, \log \left (b x + a\right )\right )}}{2 \, {\left (a p + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2)^p*(b*x+a)^(-3-2*p),x, algorithm="giac")

[Out]

1/2*((c*x^2)^p*b*x^3*e^(-2*p*log(b*x + a) - 3*log(b*x + a)) + (c*x^2)^p*a*x^2*e^(-2*p*log(b*x + a) - 3*log(b*x
+ a)))/(a*p + a)

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Mupad [B]
time = 0.22, size = 33, normalized size = 1.00 \begin {gather*} \frac {x^2\,{\left (c\,x^2\right )}^p}{2\,a\,\left (p+1\right )\,{\left (a+b\,x\right )}^{2\,p+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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