3.1059 \(\int (c x)^m (a+b x^2)^p \, dx\)

Optimal. Leaf size=66 \[ \frac {(c x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {b x^2}{a}\right )}{c (m+1)} \]

[Out]

(c*x)^(1+m)*(b*x^2+a)^p*hypergeom([-p, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/c/(1+m)/((1+b*x^2/a)^p)

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Rubi [A]  time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac {(c x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {b x^2}{a}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(a + b*x^2)^p,x]

[Out]

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int (c x)^m \left (a+b x^2\right )^p \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int (c x)^m \left (1+\frac {b x^2}{a}\right )^p \, dx\\ &=\frac {(c x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};-\frac {b x^2}{a}\right )}{c (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 64, normalized size = 0.97 \[ \frac {x (c x)^m \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+1}{2}+1;-\frac {b x^2}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(a + b*x^2)^p,x]

[Out]

(x*(c*x)^m*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*(1 + (b*x^2)/
a)^p)

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} + a\right )}^{p} \left (c x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2 + a)^p*(c*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2 + a)^p*(c*x)^m, x)

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{m} \left (b \,x^{2}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2 + a)^p*(c*x)^m, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (c\,x\right )}^m\,{\left (b\,x^2+a\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(a + b*x^2)^p,x)

[Out]

int((c*x)^m*(a + b*x^2)^p, x)

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sympy [C]  time = 21.46, size = 54, normalized size = 0.82 \[ \frac {a^{p} c^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**p*c**m*x*x**m*gamma(m/2 + 1/2)*hyper((-p, m/2 + 1/2), (m/2 + 3/2,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(m/2
+ 3/2))

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