3.46 \(\int (e x)^m (a+b x^2)^p (A+B x^2) (c+d x^2) \, dx\)

Optimal. Leaf size=253 \[ -\frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (m+2 p+3) (a d (m+1)-b c (m+2 p+5))-a (m+1) (a B d (m+3)-b (2 A d+B c (m+2 p+5))))}{b^2 e (m+1) (m+2 p+3) (m+2 p+5)}-\frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b^2 e (m+2 p+3) (m+2 p+5)}+\frac{d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)} \]

[Out]

-(((a*B*d*(3 + m) - b*(2*A*d + B*c*(5 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b^2*e*(3 + m + 2*p)*(5
+ m + 2*p))) + (d*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(A + B*x^2))/(b*e*(5 + m + 2*p)) - ((A*b*(3 + m + 2*p)*(a*
d*(1 + m) - b*c*(5 + m + 2*p)) - a*(1 + m)*(a*B*d*(3 + m) - b*(2*A*d + B*c*(5 + m + 2*p))))*(e*x)^(1 + m)*(a +
b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b^2*e*(1 + m)*(3 + m + 2*p)*(5 + m + 2*p
)*(1 + (b*x^2)/a)^p)

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Rubi [A]  time = 0.226954, antiderivative size = 238, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {581, 459, 365, 364} \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+3)+2 A b d+b B c (m+2 p+5))}{b^2 e (m+2 p+3) (m+2 p+5)}-\frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) \left (\frac{a (-a B d (m+3)+2 A b d+b B c (m+2 p+5))}{b (m+2 p+3)}+a A d-\frac{A b c (m+2 p+5)}{m+1}\right )}{b e (m+2 p+5)}+\frac{d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2),x]

[Out]

((2*A*b*d - a*B*d*(3 + m) + b*B*c*(5 + m + 2*p))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b^2*e*(3 + m + 2*p)*(5 +
m + 2*p)) + (d*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(A + B*x^2))/(b*e*(5 + m + 2*p)) - ((a*A*d - (A*b*c*(5 + m +
2*p))/(1 + m) + (a*(2*A*b*d - a*B*d*(3 + m) + b*B*c*(5 + m + 2*p)))/(b*(3 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^
2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b*e*(5 + m + 2*p)*(1 + (b*x^2)/a)^p)

Rule 581

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*g*(m + n*(p + q + 1) + 1)), x] + Dis
t[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b
*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 459

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

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])

Rubi steps

\begin{align*} \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx &=\frac{d (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (A+B x^2\right )}{b e (5+m+2 p)}+\frac{\int (e x)^m \left (a+b x^2\right )^p \left (-A (a d (1+m)-b c (5+m+2 p))+(2 A b d-a B d (3+m)+b B c (5+m+2 p)) x^2\right ) \, dx}{b (5+m+2 p)}\\ &=\frac{(2 A b d-a B d (3+m)+b B c (5+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^2 e (3+m+2 p) (5+m+2 p)}+\frac{d (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (A+B x^2\right )}{b e (5+m+2 p)}-\frac{\left (A (a d (1+m)-b c (5+m+2 p))+\frac{a (1+m) (2 A b d-a B d (3+m)+b B c (5+m+2 p))}{b (3+m+2 p)}\right ) \int (e x)^m \left (a+b x^2\right )^p \, dx}{b (5+m+2 p)}\\ &=\frac{(2 A b d-a B d (3+m)+b B c (5+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^2 e (3+m+2 p) (5+m+2 p)}+\frac{d (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (A+B x^2\right )}{b e (5+m+2 p)}-\frac{\left (\left (A (a d (1+m)-b c (5+m+2 p))+\frac{a (1+m) (2 A b d-a B d (3+m)+b B c (5+m+2 p))}{b (3+m+2 p)}\right ) \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac{b x^2}{a}\right )^p \, dx}{b (5+m+2 p)}\\ &=\frac{(2 A b d-a B d (3+m)+b B c (5+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^2 e (3+m+2 p) (5+m+2 p)}+\frac{d (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (A+B x^2\right )}{b e (5+m+2 p)}-\frac{\left (A (a d (1+m)-b c (5+m+2 p))+\frac{a (1+m) (2 A b d-a B d (3+m)+b B c (5+m+2 p))}{b (3+m+2 p)}\right ) (e 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 )}{b e (1+m) (5+m+2 p)}\\ \end{align*}

Mathematica [A]  time = 0.117018, size = 147, normalized size = 0.58 \[ x (e x)^m \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (\frac{x^2 (A d+B c) \, _2F_1\left (\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{b x^2}{a}\right )}{m+3}+\frac{A c \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right )}{m+1}+\frac{B d x^4 \, _2F_1\left (\frac{m+5}{2},-p;\frac{m+7}{2};-\frac{b x^2}{a}\right )}{m+5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2),x]

[Out]

(x*(e*x)^m*(a + b*x^2)^p*((A*c*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(1 + m) + ((B*c + A*
d)*x^2*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + (B*d*x^4*Hypergeometric2F1[(5 + m)
/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m)))/(1 + (b*x^2)/a)^p

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{p} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c),x)

[Out]

int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x^2 + A)*(d*x^2 + c)*(b*x^2 + a)^p*(e*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B d x^{4} +{\left (B c + A d\right )} x^{2} + A c\right )}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*d*x^4 + (B*c + A*d)*x^2 + A*c)*(b*x^2 + a)^p*(e*x)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(b*x**2+a)**p*(B*x**2+A)*(d*x**2+c),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x^2 + A)*(d*x^2 + c)*(b*x^2 + a)^p*(e*x)^m, x)