3.1152 \(\int (e x)^{3/2} \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\)

Optimal. Leaf size=91 \[ \frac{2 (e x)^{5/2} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{5}{4};-p,-q;\frac{9}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{5 e} \]

[Out]

(2*(e*x)^(5/2)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[5/4, -p, -q, 9/4, -((b*x^2)/
a), -((d*x^2)/c)])/(5*e*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)

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Rubi [A]  time = 0.217305, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 (e x)^{5/2} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{5}{4};-p,-q;\frac{9}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{5 e} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(3/2)*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

(2*(e*x)^(5/2)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[5/4, -p, -q, 9/4, -((b*x^2)/
a), -((d*x^2)/c)])/(5*e*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)

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Rubi in Sympy [A]  time = 33.0135, size = 71, normalized size = 0.78 \[ \frac{2 \left (e x\right )^{\frac{5}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (1 + \frac{d x^{2}}{c}\right )^{- q} \left (a + b x^{2}\right )^{p} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{5}{4},- p,- q,\frac{9}{4},- \frac{b x^{2}}{a},- \frac{d x^{2}}{c} \right )}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(3/2)*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

2*(e*x)**(5/2)*(1 + b*x**2/a)**(-p)*(1 + d*x**2/c)**(-q)*(a + b*x**2)**p*(c + d*
x**2)**q*appellf1(5/4, -p, -q, 9/4, -b*x**2/a, -d*x**2/c)/(5*e)

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Mathematica [A]  time = 0.365757, size = 181, normalized size = 1.99 \[ \frac{18 a c x (e x)^{3/2} \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (\frac{5}{4};-p,-q;\frac{9}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{5 \left (4 x^2 \left (b c p F_1\left (\frac{9}{4};1-p,-q;\frac{13}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (\frac{9}{4};-p,1-q;\frac{13}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+9 a c F_1\left (\frac{5}{4};-p,-q;\frac{9}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(3/2)*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

(18*a*c*x*(e*x)^(3/2)*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[5/4, -p, -q, 9/4, -((
b*x^2)/a), -((d*x^2)/c)])/(5*(9*a*c*AppellF1[5/4, -p, -q, 9/4, -((b*x^2)/a), -((
d*x^2)/c)] + 4*x^2*(b*c*p*AppellF1[9/4, 1 - p, -q, 13/4, -((b*x^2)/a), -((d*x^2)
/c)] + a*d*q*AppellF1[9/4, -p, 1 - q, 13/4, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(3/2)*(b*x^2+a)^p*(d*x^2+c)^q,x)

[Out]

int((e*x)^(3/2)*(b*x^2+a)^p*(d*x^2+c)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (e x\right )^{\frac{3}{2}}{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(3/2)*(b*x^2 + a)^p*(d*x^2 + c)^q,x, algorithm="maxima")

[Out]

integrate((e*x)^(3/2)*(b*x^2 + a)^p*(d*x^2 + c)^q, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(3/2)*(b*x^2 + a)^p*(d*x^2 + c)^q,x, algorithm="fricas")

[Out]

integral(sqrt(e*x)*(b*x^2 + a)^p*(d*x^2 + c)^q*e*x, 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)**(3/2)*(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. \[ \int \left (e x\right )^{\frac{3}{2}}{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(3/2)*(b*x^2 + a)^p*(d*x^2 + c)^q,x, algorithm="giac")

[Out]

integrate((e*x)^(3/2)*(b*x^2 + a)^p*(d*x^2 + c)^q, x)