3.1154 \(\int \frac{\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{\sqrt{e x}} \, dx\)

Optimal. Leaf size=89 \[ \frac{2 \sqrt{e x} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.216459, antiderivative size = 89, 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 \[ \frac{2 \sqrt{e x} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.1572, size = 70, normalized size = 0.79 \[ \frac{2 \sqrt{e x} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (1 + \frac{d x^{2}}{c}\right )^{- q} \left (a + b x^{2}\right )^{p} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{4},- p,- q,\frac{5}{4},- \frac{b x^{2}}{a},- \frac{d x^{2}}{c} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**p*(d*x**2+c)**q/(e*x)**(1/2),x)

[Out]

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

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Mathematica [B]  time = 0.374481, size = 179, normalized size = 2.01 \[ \frac{10 a c x \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\sqrt{e x} \left (4 x^2 \left (b c p F_1\left (\frac{5}{4};1-p,-q;\frac{9}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (\frac{5}{4};-p,1-q;\frac{9}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}{\frac{1}{\sqrt{ex}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^p*(d*x^2+c)^q/(e*x)^(1/2),x)

[Out]

int((b*x^2+a)^p*(d*x^2+c)^q/(e*x)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^p*(d*x^2 + c)^q/sqrt(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((b*x**2+a)**p*(d*x**2+c)**q/(e*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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