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

Optimal. Leaf size=132 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (a d (1-m)+b c (m+1)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{2 c d e (m+1) \sqrt{c+d x^4}}-\frac{(e x)^{m+1} (b c-a d)}{2 c d e \sqrt{c+d x^4}} \]

[Out]

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

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Rubi [A]  time = 0.188099, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (a d (1-m)+b c (m+1)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{2 c d e (m+1) \sqrt{c+d x^4}}-\frac{(e x)^{m+1} (b c-a d)}{2 c d e \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.1395, size = 105, normalized size = 0.8 \[ \frac{\left (e x\right )^{m + 1} \left (a d - b c\right )}{2 c d e \sqrt{c + d x^{4}}} + \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \left (a d \left (- m + 1\right ) + b c \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{2 c^{2} d e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.132366, size = 110, normalized size = 0.83 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left ((a d-b c) \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )+b c \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )\right )}{c d (m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(b*c*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4,
-((d*x^4)/c)] + (-(b*c) + a*d)*Hypergeometric2F1[3/2, (1 + m)/4, (5 + m)/4, -((
d*x^4)/c)]))/(c*d*(1 + m)*Sqrt[c + d*x^4])

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( b{x}^{4}+a \right ) \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 178.942, size = 119, normalized size = 0.9 \[ \frac{a e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 c^{\frac{3}{2}} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} + \frac{b e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{4} + \frac{5}{4} \\ \frac{m}{4} + \frac{9}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 c^{\frac{3}{2}} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*e**m*x*x**m*gamma(m/4 + 1/4)*hyper((3/2, m/4 + 1/4), (m/4 + 5/4,), d*x**4*exp_
polar(I*pi)/c)/(4*c**(3/2)*gamma(m/4 + 5/4)) + b*e**m*x**5*x**m*gamma(m/4 + 5/4)
*hyper((3/2, m/4 + 5/4), (m/4 + 9/4,), d*x**4*exp_polar(I*pi)/c)/(4*c**(3/2)*gam
ma(m/4 + 9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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