3.10.23 \(\int (c x^2)^p (a+b x)^{-2-2 p} \, dx\)

Optimal. Leaf size=30 \[ \frac {x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{a (2 p+1)} \]

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {15, 37} \begin {gather*} \frac {x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{a (2 p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^p*(a + b*x)^(-2 - 2*p),x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.93 \begin {gather*} \frac {x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{2 a p+a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c x^2\right )^p (a+b x)^{-2-2 p} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][(c*x^2)^p*(a + b*x)^(-2 - 2*p), x]

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fricas [A]  time = 1.37, size = 36, normalized size = 1.20 \begin {gather*} \frac {{\left (b x^{2} + a x\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 2}}{2 \, a p + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 31, normalized size = 1.03 \begin {gather*} \frac {x \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-2 p -1}}{\left (2 p +1\right ) a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.20, size = 32, normalized size = 1.07 \begin {gather*} \frac {x\,{\left (c\,x^2\right )}^p}{a\,\left (2\,p+1\right )\,{\left (a+b\,x\right )}^{2\,p+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^p/(a + b*x)^(2*p + 2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} - \frac {b^{- 2 p} c^{p} x^{- 2 p} \left (x^{2}\right )^{p}}{b^{2} x} & \text {for}\: a = 0 \\\frac {0^{- 2 p - 2} c^{p} x \left (x^{2}\right )^{p}}{2 p + 1} & \text {for}\: a = - b x \\\frac {c^{p} x \left (0^{\frac {1}{p}}\right )^{- 2 p - 2} \left (x^{2}\right )^{p}}{2 p + 1} & \text {for}\: a = 0^{\frac {1}{p}} - b x \\\int \frac {1}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a^{3} c^{p} x \left (x^{2}\right )^{p}}{2 a^{5} p \left (a + b x\right )^{2 p} + a^{5} \left (a + b x\right )^{2 p} + 8 a^{4} b p x \left (a + b x\right )^{2 p} + 4 a^{4} b x \left (a + b x\right )^{2 p} + 12 a^{3} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 6 a^{3} b^{2} x^{2} \left (a + b x\right )^{2 p} + 8 a^{2} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 4 a^{2} b^{3} x^{3} \left (a + b x\right )^{2 p} + 2 a b^{4} p x^{4} \left (a + b x\right )^{2 p} + a b^{4} x^{4} \left (a + b x\right )^{2 p}} + \frac {2 a^{2} b c^{p} x^{2} \left (x^{2}\right )^{p}}{2 a^{5} p \left (a + b x\right )^{2 p} + a^{5} \left (a + b x\right )^{2 p} + 8 a^{4} b p x \left (a + b x\right )^{2 p} + 4 a^{4} b x \left (a + b x\right )^{2 p} + 12 a^{3} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 6 a^{3} b^{2} x^{2} \left (a + b x\right )^{2 p} + 8 a^{2} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 4 a^{2} b^{3} x^{3} \left (a + b x\right )^{2 p} + 2 a b^{4} p x^{4} \left (a + b x\right )^{2 p} + a b^{4} x^{4} \left (a + b x\right )^{2 p}} + \frac {a b^{2} c^{p} x^{3} \left (x^{2}\right )^{p}}{2 a^{5} p \left (a + b x\right )^{2 p} + a^{5} \left (a + b x\right )^{2 p} + 8 a^{4} b p x \left (a + b x\right )^{2 p} + 4 a^{4} b x \left (a + b x\right )^{2 p} + 12 a^{3} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 6 a^{3} b^{2} x^{2} \left (a + b x\right )^{2 p} + 8 a^{2} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 4 a^{2} b^{3} x^{3} \left (a + b x\right )^{2 p} + 2 a b^{4} p x^{4} \left (a + b x\right )^{2 p} + a b^{4} x^{4} \left (a + b x\right )^{2 p}} + \frac {b c^{p} x^{2} \left (x^{2}\right )^{p}}{2 a^{3} p \left (a + b x\right )^{2 p} + a^{3} \left (a + b x\right )^{2 p} + 4 a^{2} b p x \left (a + b x\right )^{2 p} + 2 a^{2} b x \left (a + b x\right )^{2 p} + 2 a b^{2} p x^{2} \left (a + b x\right )^{2 p} + a b^{2} x^{2} \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((c*x**2)**p*(b*x+a)**(-2-2*p),x)

[Out]

Piecewise((-b**(-2*p)*c**p*x**(-2*p)*(x**2)**p/(b**2*x), Eq(a, 0)), (0**(-2*p - 2)*c**p*x*(x**2)**p/(2*p + 1),
Eq(a, -b*x)), (c**p*x*(0**(1/p))**(-2*p - 2)*(x**2)**p/(2*p + 1), Eq(a, 0**(1/p) - b*x)), (Integral(1/(sqrt(c
*x**2)*(a + b*x)), x), Eq(p, -1/2)), (a**3*c**p*x*(x**2)**p/(2*a**5*p*(a + b*x)**(2*p) + a**5*(a + b*x)**(2*p)
+ 8*a**4*b*p*x*(a + b*x)**(2*p) + 4*a**4*b*x*(a + b*x)**(2*p) + 12*a**3*b**2*p*x**2*(a + b*x)**(2*p) + 6*a**3
*b**2*x**2*(a + b*x)**(2*p) + 8*a**2*b**3*p*x**3*(a + b*x)**(2*p) + 4*a**2*b**3*x**3*(a + b*x)**(2*p) + 2*a*b*
*4*p*x**4*(a + b*x)**(2*p) + a*b**4*x**4*(a + b*x)**(2*p)) + 2*a**2*b*c**p*x**2*(x**2)**p/(2*a**5*p*(a + b*x)*
*(2*p) + a**5*(a + b*x)**(2*p) + 8*a**4*b*p*x*(a + b*x)**(2*p) + 4*a**4*b*x*(a + b*x)**(2*p) + 12*a**3*b**2*p*
x**2*(a + b*x)**(2*p) + 6*a**3*b**2*x**2*(a + b*x)**(2*p) + 8*a**2*b**3*p*x**3*(a + b*x)**(2*p) + 4*a**2*b**3*
x**3*(a + b*x)**(2*p) + 2*a*b**4*p*x**4*(a + b*x)**(2*p) + a*b**4*x**4*(a + b*x)**(2*p)) + a*b**2*c**p*x**3*(x
**2)**p/(2*a**5*p*(a + b*x)**(2*p) + a**5*(a + b*x)**(2*p) + 8*a**4*b*p*x*(a + b*x)**(2*p) + 4*a**4*b*x*(a + b
*x)**(2*p) + 12*a**3*b**2*p*x**2*(a + b*x)**(2*p) + 6*a**3*b**2*x**2*(a + b*x)**(2*p) + 8*a**2*b**3*p*x**3*(a
+ b*x)**(2*p) + 4*a**2*b**3*x**3*(a + b*x)**(2*p) + 2*a*b**4*p*x**4*(a + b*x)**(2*p) + a*b**4*x**4*(a + b*x)**
(2*p)) + b*c**p*x**2*(x**2)**p/(2*a**3*p*(a + b*x)**(2*p) + a**3*(a + b*x)**(2*p) + 4*a**2*b*p*x*(a + b*x)**(2
*p) + 2*a**2*b*x*(a + b*x)**(2*p) + 2*a*b**2*p*x**2*(a + b*x)**(2*p) + a*b**2*x**2*(a + b*x)**(2*p)), True))

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