3.683 \(\int \frac{(e x)^m}{\left (a+b x^4\right )^3 \left (c+d x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^3 c e (m+1) \sqrt{c+d x^4}} \]

[Out]

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

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Rubi [A]  time = 0.208167, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^3 c e (m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m/((a + b*x^4)^3*(c + d*x^4)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.5962, size = 68, normalized size = 0.81 \[ \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{4} + \frac{1}{4},\frac{3}{2},3,\frac{m}{4} + \frac{5}{4},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{a^{3} c^{2} e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m/(b*x**4+a)**3/(d*x**4+c)**(3/2),x)

[Out]

(e*x)**(m + 1)*sqrt(c + d*x**4)*appellf1(m/4 + 1/4, 3/2, 3, m/4 + 5/4, -d*x**4/c
, -b*x**4/a)/(a**3*c**2*e*sqrt(1 + d*x**4/c)*(m + 1))

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Mathematica [B]  time = 0.736128, size = 209, normalized size = 2.49 \[ -\frac{a c (m+5) x (e x)^m F_1\left (\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{(m+1) \left (a+b x^4\right )^3 \left (c+d x^4\right )^{3/2} \left (6 x^4 \left (a d F_1\left (\frac{m+5}{4};3,\frac{5}{2};\frac{m+9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+2 b c F_1\left (\frac{m+5}{4};4,\frac{3}{2};\frac{m+9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-a c (m+5) F_1\left (\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^m/((a + b*x^4)^3*(c + d*x^4)^(3/2)),x]

[Out]

-((a*c*(5 + m)*x*(e*x)^m*AppellF1[(1 + m)/4, 3, 3/2, (5 + m)/4, -((b*x^4)/a), -(
(d*x^4)/c)])/((1 + m)*(a + b*x^4)^3*(c + d*x^4)^(3/2)*(-(a*c*(5 + m)*AppellF1[(1
+ m)/4, 3, 3/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)]) + 6*x^4*(a*d*AppellF1[(
5 + m)/4, 3, 5/2, (9 + m)/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*b*c*AppellF1[(5 + m
)/4, 4, 3/2, (9 + m)/4, -((b*x^4)/a), -((d*x^4)/c)]))))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( b{x}^{4}+a \right ) ^{3}} \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m/(b*x^4+a)^3/(d*x^4+c)^(3/2),x)

[Out]

int((e*x)^m/(b*x^4+a)^3/(d*x^4+c)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}^{3}{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/((b*x^4 + a)^3*(d*x^4 + c)^(3/2)),x, algorithm="maxima")

[Out]

integrate((e*x)^m/((b*x^4 + a)^3*(d*x^4 + c)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (e x\right )^{m}}{{\left (b^{3} d x^{16} +{\left (b^{3} c + 3 \, a b^{2} d\right )} x^{12} + 3 \,{\left (a b^{2} c + a^{2} b d\right )} x^{8} +{\left (3 \, a^{2} b c + a^{3} d\right )} x^{4} + a^{3} c\right )} \sqrt{d x^{4} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/((b*x^4 + a)^3*(d*x^4 + c)^(3/2)),x, algorithm="fricas")

[Out]

integral((e*x)^m/((b^3*d*x^16 + (b^3*c + 3*a*b^2*d)*x^12 + 3*(a*b^2*c + a^2*b*d)
*x^8 + (3*a^2*b*c + a^3*d)*x^4 + a^3*c)*sqrt(d*x^4 + c)), 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**4+a)**3/(d*x**4+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )}^{3}{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/((b*x^4 + a)^3*(d*x^4 + c)^(3/2)),x, algorithm="giac")

[Out]

integrate((e*x)^m/((b*x^4 + a)^3*(d*x^4 + c)^(3/2)), x)