3.1139 \(\int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\)

Optimal. Leaf size=101 \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e (m+1)} \]

[Out]

((e*x)^(1 + m)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[(1 + m)/2, -p, -q, (3 + m)/2
, -((b*x^2)/a), -((d*x^2)/c)])/(e*(1 + m)*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)

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Rubi [A]  time = 0.234007, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

((e*x)^(1 + m)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[(1 + m)/2, -p, -q, (3 + m)/2
, -((b*x^2)/a), -((d*x^2)/c)])/(e*(1 + m)*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)

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Rubi in Sympy [A]  time = 32.7982, size = 78, normalized size = 0.77 \[ \frac{\left (e x\right )^{m + 1} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (1 + \frac{d x^{2}}{c}\right )^{- q} \left (a + b x^{2}\right )^{p} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{1}{2},- p,- q,\frac{m}{2} + \frac{3}{2},- \frac{b x^{2}}{a},- \frac{d x^{2}}{c} \right )}}{e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

(e*x)**(m + 1)*(1 + b*x**2/a)**(-p)*(1 + d*x**2/c)**(-q)*(a + b*x**2)**p*(c + d*
x**2)**q*appellf1(m/2 + 1/2, -p, -q, m/2 + 3/2, -b*x**2/a, -d*x**2/c)/(e*(m + 1)
)

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Mathematica [B]  time = 0.606023, size = 218, normalized size = 2.16 \[ \frac{a c (m+3) x (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{(m+1) \left (2 x^2 \left (b c p F_1\left (\frac{m+3}{2};1-p,-q;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (\frac{m+3}{2};-p,1-q;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+a c (m+3) F_1\left (\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*c*(3 + m)*x*(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[(1 + m)/2, -p, -q, (
3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)])/((1 + m)*(a*c*(3 + m)*AppellF1[(1 + m)/2,
-p, -q, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x^2*(b*c*p*AppellF1[(3 + m)/
2, 1 - p, -q, (5 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] + a*d*q*AppellF1[(3 + m)/2,
-p, 1 - q, (5 + m)/2, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.206, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(b*x^2+a)^p*(d*x^2+c)^q,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*(e*x)^m,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} \left (e x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*(e*x)^m,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} \left (e x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*(e*x)^m,x, algorithm="giac")

[Out]

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