3.1156 \(\int \frac{\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{(e x)^{5/2}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{2 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 e (e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.220991, 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 \[ -\frac{2 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 e (e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^p*(c + d*x^2)^q)/(e*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 33.0525, size = 75, normalized size = 0.82 \[ - \frac{2 \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (1 + \frac{d x^{2}}{c}\right )^{- q} \left (a + b x^{2}\right )^{p} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (- \frac{3}{4},- p,- q,\frac{1}{4},- \frac{b x^{2}}{a},- \frac{d x^{2}}{c} \right )}}{3 e \left (e x\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

-2*(1 + b*x**2/a)**(-p)*(1 + d*x**2/c)**(-q)*(a + b*x**2)**p*(c + d*x**2)**q*app
ellf1(-3/4, -p, -q, 1/4, -b*x**2/a, -d*x**2/c)/(3*e*(e*x)**(3/2))

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Mathematica [A]  time = 0.383027, size = 180, normalized size = 1.98 \[ -\frac{2 a c x \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 (e x)^{5/2} \left (4 x^2 \left (b c p F_1\left (\frac{1}{4};1-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (\frac{1}{4};-p,1-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+a c F_1\left (-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x^2)^p*(c + d*x^2)^q)/(e*x)^(5/2),x]

[Out]

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

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q} \left ( ex \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^p*(d*x^2+c)^q/(e*x)^(5/2),x)

[Out]

int((b*x^2+a)^p*(d*x^2+c)^q/(e*x)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/(e*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q/(e*x)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/(e*x)^(5/2),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p*(d*x^2 + c)^q/(sqrt(e*x)*e^2*x^2), 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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q/(e*x)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q/(e*x)^(5/2), x)