3.675 \(\int \frac{(e x)^m}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=81 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};1,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e (m+1) \sqrt{c+d x^4}} \]

[Out]

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

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Rubi [A]  time = 0.19969, antiderivative size = 81, 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 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};1,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e (m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.314, size = 65, normalized size = 0.8 \[ \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{4} + \frac{1}{4},\frac{1}{2},1,\frac{m}{4} + \frac{5}{4},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{a c e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

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

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Mathematica [B]  time = 0.614377, size = 282, normalized size = 3.48 \[ \frac{x (e x)^m \left (-\frac{a b c (m+5) \left (c+d x^4\right ) F_1\left (\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) (a d-b c) \left (2 x^4 \left (a d F_1\left (\frac{m+5}{4};\frac{1}{2},1;\frac{m+9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-2 b c F_1\left (\frac{m+5}{4};-\frac{1}{2},2;\frac{m+9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )+a c (m+5) F_1\left (\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{d \sqrt{\frac{d x^4}{c}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{b c-a d}\right )}{(m+1) \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(e*x)^m*(-((a*b*c*(5 + m)*(c + d*x^4)*AppellF1[(1 + m)/4, -1/2, 1, (5 + m)/4,
-((d*x^4)/c), -((b*x^4)/a)])/((-(b*c) + a*d)*(a + b*x^4)*(a*c*(5 + m)*AppellF1[
(1 + m)/4, -1/2, 1, (5 + m)/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(-2*b*c*Appel
lF1[(5 + m)/4, -1/2, 2, (9 + m)/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[(5
+ m)/4, 1/2, 1, (9 + m)/4, -((d*x^4)/c), -((b*x^4)/a)])))) - (d*Sqrt[1 + (d*x^4
)/c]*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)])/(b*c - a*d)))/(
(1 + m)*Sqrt[c + d*x^4])

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{b{x}^{4}+a}{\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

int((e*x)^m/(b*x^4+a)/(d*x^4+c)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x)^m/((b*x^4 + a)*sqrt(d*x^4 + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral((e*x)**m/((a + b*x**4)*sqrt(c + d*x**4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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