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))
________________________________________________________________________________________
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, 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))
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 141.62, size = 1406, normalized size = 42.61
result too large to display
Warning: Unable to verify antiderivative.
[In]
mathics('Integrate[x^1*(c*x^2)^p/(a + b*x)^(2*p + 3),x]')
[Out]
Piecewise[{{-(b x) ^ (-2 p) (c x ^ 2) ^ p / (b ^ 3 x), a == 0}, {x ^ 2 (c x ^ 2) ^ p ComplexInfinity ^ (1 + 2
p) / (1 + p), a == -b x}, {x ^ 2 (c x ^ 2) ^ p (0 ^ (1 / p)) ^ (-3 - 2 p) / (2 (1 + p)), a == -b x + 0 ^ (1 /
p)}, {(Log[x] - Log[a / b + x]) / (a c), p == -1}}, a ^ 4 x ^ 2 (c x ^ 2) ^ p / (2 a ^ 7 (a + b x) ^ (2 p) + 2
a ^ 7 p (a + b x) ^ (2 p) + 12 a ^ 6 b x (a + b x) ^ (2 p) + 12 a ^ 6 b p x (a + b x) ^ (2 p) + 30 a ^ 5 b ^
2 x ^ 2 (a + b x) ^ (2 p) + 30 a ^ 5 b ^ 2 p x ^ 2 (a + b x) ^ (2 p) + 40 a ^ 4 b ^ 3 x ^ 3 (a + b x) ^ (2 p)
+ 40 a ^ 4 b ^ 3 p x ^ 3 (a + b x) ^ (2 p) + 30 a ^ 3 b ^ 4 x ^ 4 (a + b x) ^ (2 p) + 30 a ^ 3 b ^ 4 p x ^ 4 (
a + b x) ^ (2 p) + 12 a ^ 2 b ^ 5 x ^ 5 (a + b x) ^ (2 p) + 12 a ^ 2 b ^ 5 p x ^ 5 (a + b x) ^ (2 p) + 2 a b ^
6 x ^ 6 (a + b x) ^ (2 p) + 2 a b ^ 6 p x ^ 6 (a + b x) ^ (2 p)) + b x ^ 3 (c x ^ 2) ^ p / (2 a ^ 4 (a + b x)
^ (2 p) + 2 a ^ 4 p (a + b x) ^ (2 p) + 6 a ^ 3 b x (a + b x) ^ (2 p) + 6 a ^ 3 b p x (a + b x) ^ (2 p) + 6 a
^ 2 b ^ 2 x ^ 2 (a + b x) ^ (2 p) + 6 a ^ 2 b ^ 2 p x ^ 2 (a + b x) ^ (2 p) + 2 a b ^ 3 x ^ 3 (a + b x) ^ (2
p) + 2 a b ^ 3 p x ^ 3 (a + b x) ^ (2 p)) + 3 a ^ 3 b x ^ 3 (c x ^ 2) ^ p / (2 a ^ 7 (a + b x) ^ (2 p) + 2 a ^
7 p (a + b x) ^ (2 p) + 12 a ^ 6 b x (a + b x) ^ (2 p) + 12 a ^ 6 b p x (a + b x) ^ (2 p) + 30 a ^ 5 b ^ 2 x
^ 2 (a + b x) ^ (2 p) + 30 a ^ 5 b ^ 2 p x ^ 2 (a + b x) ^ (2 p) + 40 a ^ 4 b ^ 3 x ^ 3 (a + b x) ^ (2 p) + 40
a ^ 4 b ^ 3 p x ^ 3 (a + b x) ^ (2 p) + 30 a ^ 3 b ^ 4 x ^ 4 (a + b x) ^ (2 p) + 30 a ^ 3 b ^ 4 p x ^ 4 (a +
b x) ^ (2 p) + 12 a ^ 2 b ^ 5 x ^ 5 (a + b x) ^ (2 p) + 12 a ^ 2 b ^ 5 p x ^ 5 (a + b x) ^ (2 p) + 2 a b ^ 6 x
^ 6 (a + b x) ^ (2 p) + 2 a b ^ 6 p x ^ 6 (a + b x) ^ (2 p)) + 3 a ^ 2 b ^ 2 x ^ 4 (c x ^ 2) ^ p / (2 a ^ 7 (
a + b x) ^ (2 p) + 2 a ^ 7 p (a + b x) ^ (2 p) + 12 a ^ 6 b x (a + b x) ^ (2 p) + 12 a ^ 6 b p x (a + b x) ^ (
2 p) + 30 a ^ 5 b ^ 2 x ^ 2 (a + b x) ^ (2 p) + 30 a ^ 5 b ^ 2 p x ^ 2 (a + b x) ^ (2 p) + 40 a ^ 4 b ^ 3 x ^
3 (a + b x) ^ (2 p) + 40 a ^ 4 b ^ 3 p x ^ 3 (a + b x) ^ (2 p) + 30 a ^ 3 b ^ 4 x ^ 4 (a + b x) ^ (2 p) + 30 a
^ 3 b ^ 4 p x ^ 4 (a + b x) ^ (2 p) + 12 a ^ 2 b ^ 5 x ^ 5 (a + b x) ^ (2 p) + 12 a ^ 2 b ^ 5 p x ^ 5 (a + b
x) ^ (2 p) + 2 a b ^ 6 x ^ 6 (a + b x) ^ (2 p) + 2 a b ^ 6 p x ^ 6 (a + b x) ^ (2 p)) + a b ^ 3 x ^ 5 (c x ^ 2
) ^ p / (2 a ^ 7 (a + b x) ^ (2 p) + 2 a ^ 7 p (a + b x) ^ (2 p) + 12 a ^ 6 b x (a + b x) ^ (2 p) + 12 a ^ 6 b
p x (a + b x) ^ (2 p) + 30 a ^ 5 b ^ 2 x ^ 2 (a + b x) ^ (2 p) + 30 a ^ 5 b ^ 2 p x ^ 2 (a + b x) ^ (2 p) + 4
0 a ^ 4 b ^ 3 x ^ 3 (a + b x) ^ (2 p) + 40 a ^ 4 b ^ 3 p x ^ 3 (a + b x) ^ (2 p) + 30 a ^ 3 b ^ 4 x ^ 4 (a + b
x) ^ (2 p) + 30 a ^ 3 b ^ 4 p x ^ 4 (a + b x) ^ (2 p) + 12 a ^ 2 b ^ 5 x ^ 5 (a + b x) ^ (2 p) + 12 a ^ 2 b ^
5 p x ^ 5 (a + b x) ^ (2 p) + 2 a b ^ 6 x ^ 6 (a + b x) ^ (2 p) + 2 a b ^ 6 p x ^ 6 (a + b x) ^ (2 p))]
________________________________________________________________________________________
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
________________________________________________________________________________________
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)
________________________________________________________________________________________
Fricas [A]
time = 0.30, 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)
________________________________________________________________________________________
Sympy [A]
time = 147.57, size = 1409, normalized size = 42.70
result too large to display
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))
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs.
\(2 (33) = 66\).
time = 0.01, size = 81, normalized size = 2.45 \begin {gather*} \frac {a x^{2} \mathrm {e}^{-2 p \ln \left (a+b x\right )-3 \ln \left (a+b x\right )} \mathrm {e}^{p \ln \left (c x^{2}\right )}+b x^{3} \mathrm {e}^{-2 p \ln \left (a+b x\right )-3 \ln \left (a+b x\right )} \mathrm {e}^{p \ln \left (c x^{2}\right )}}{2 a p+2 a} \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]
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)
________________________________________________________________________________________
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))
________________________________________________________________________________________