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

Optimal. Leaf size=123 \[ \frac{b \sqrt{c+d x^4} (e x)^{m+1}}{d e (m+3)}-\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (b c (m+1)-a d (m+3)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{d e (m+1) (m+3) \sqrt{c+d x^4}} \]

[Out]

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

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Rubi [A]  time = 0.183351, antiderivative size = 115, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (\frac{a}{m+1}-\frac{b c}{d (m+3)}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{e \sqrt{c+d x^4}}+\frac{b \sqrt{c+d x^4} (e x)^{m+1}}{d e (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.3996, size = 100, normalized size = 0.81 \[ \frac{b \left (e x\right )^{m + 1} \sqrt{c + d x^{4}}}{d e \left (m + 3\right )} + \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \left (a d \left (m + 3\right ) - b c \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{c d e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right ) \left (m + 3\right )} \]

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]

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

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Mathematica [A]  time = 0.133493, size = 110, normalized size = 0.89 \[ \frac{x \sqrt{c+d x^4} (e x)^m \left ((a d-b c) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )+b c \, _2F_1\left (-\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )\right )}{c d (m+1) \sqrt{\frac{d x^4}{c}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( b{x}^{4}+a \right ){\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.5919, size = 119, normalized size = 0.97 \[ \frac{a e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} + \frac{b e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{5}{4} \\ \frac{m}{4} + \frac{9}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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