3.1140 \(\int x^4 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\)

Optimal. Leaf size=84 \[ \frac{1}{5} x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{5}{2};-p,-q;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right ) \]

[Out]

(x^5*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[5/2, -p, -q, 7/2, -((b*x^2)/a), -((d*x
^2)/c)])/(5*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)

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Rubi [A]  time = 0.221904, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{5} x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{5}{2};-p,-q;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

(x^5*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[5/2, -p, -q, 7/2, -((b*x^2)/a), -((d*x
^2)/c)])/(5*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)

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Rubi in Sympy [A]  time = 29.1757, size = 65, normalized size = 0.77 \[ \frac{x^{5} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (1 + \frac{d x^{2}}{c}\right )^{- q} \left (a + b x^{2}\right )^{p} \left (c + d x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{5}{2},- p,- q,\frac{7}{2},- \frac{b x^{2}}{a},- \frac{d x^{2}}{c} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

x**5*(1 + b*x**2/a)**(-p)*(1 + d*x**2/c)**(-q)*(a + b*x**2)**p*(c + d*x**2)**q*a
ppellf1(5/2, -p, -q, 7/2, -b*x**2/a, -d*x**2/c)/5

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Mathematica [B]  time = 0.359984, size = 176, normalized size = 2.1 \[ \frac{7 a c x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (\frac{5}{2};-p,-q;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{5 \left (2 x^2 \left (b c p F_1\left (\frac{7}{2};1-p,-q;\frac{9}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (\frac{7}{2};-p,1-q;\frac{9}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+7 a c F_1\left (\frac{5}{2};-p,-q;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

(7*a*c*x^5*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[5/2, -p, -q, 7/2, -((b*x^2)/a),
-((d*x^2)/c)])/(5*(7*a*c*AppellF1[5/2, -p, -q, 7/2, -((b*x^2)/a), -((d*x^2)/c)]
+ 2*x^2*(b*c*p*AppellF1[7/2, 1 - p, -q, 9/2, -((b*x^2)/a), -((d*x^2)/c)] + a*d*q
*AppellF1[7/2, -p, 1 - q, 9/2, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)^p*(d*x^2+c)^q,x)

[Out]

int(x^4*(b*x^2+a)^p*(d*x^2+c)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^4,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^4,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p*(d*x^2 + c)^q*x^4, 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(x**4*(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. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^4,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^4, x)