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

Optimal. Leaf size=200 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (a^2 d^2 (m+3) (m+7)+b c (m+1) (b c (m+5)-2 a d (m+7))\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{d^2 e (m+1) (m+3) (m+7) \sqrt{c+d x^4}}-\frac{b \sqrt{c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d^2 e (m+3) (m+7)}+\frac{b^2 \sqrt{c+d x^4} (e x)^{m+5}}{d e^5 (m+7)} \]

[Out]

-((b*(b*c*(5 + m) - 2*a*d*(7 + m))*(e*x)^(1 + m)*Sqrt[c + d*x^4])/(d^2*e*(3 + m)
*(7 + m))) + (b^2*(e*x)^(5 + m)*Sqrt[c + d*x^4])/(d*e^5*(7 + m)) + ((a^2*d^2*(3
+ m)*(7 + m) + b*c*(1 + m)*(b*c*(5 + m) - 2*a*d*(7 + 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^2*e*(
1 + m)*(3 + m)*(7 + m)*Sqrt[c + d*x^4])

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Rubi [A]  time = 0.566222, antiderivative size = 194, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (\frac{a^2 d^2 (m+7)}{m+1}+\frac{b c (b c (m+5)-2 a d (m+7))}{m+3}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{d^2 e (m+7) \sqrt{c+d x^4}}-\frac{b \sqrt{c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d^2 e (m+3) (m+7)}+\frac{b^2 \sqrt{c+d x^4} (e x)^{m+5}}{d e^5 (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(b*c*(5 + m) - 2*a*d*(7 + m))*(e*x)^(1 + m)*Sqrt[c + d*x^4])/(d^2*e*(3 + m)
*(7 + m))) + (b^2*(e*x)^(5 + m)*Sqrt[c + d*x^4])/(d*e^5*(7 + m)) + (((a^2*d^2*(7
+ m))/(1 + m) + (b*c*(b*c*(5 + m) - 2*a*d*(7 + m)))/(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^2
*e*(7 + m)*Sqrt[c + d*x^4])

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Rubi in Sympy [A]  time = 46.3791, size = 180, normalized size = 0.9 \[ \frac{b^{2} \left (e x\right )^{m + 5} \sqrt{c + d x^{4}}}{d e^{5} \left (m + 7\right )} + \frac{b \left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \left (4 a d + \left (m + 5\right ) \left (2 a d - b c\right )\right )}{d^{2} e \left (m + 3\right ) \left (m + 7\right )} + \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \left (a^{2} d^{2} \left (m + 3\right ) \left (m + 7\right ) - b c \left (m + 1\right ) \left (4 a d + \left (m + 5\right ) \left (2 a d - b c\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{c d^{2} e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right ) \left (m + 3\right ) \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*(e*x)**(m + 5)*sqrt(c + d*x**4)/(d*e**5*(m + 7)) + b*(e*x)**(m + 1)*sqrt(c
+ d*x**4)*(4*a*d + (m + 5)*(2*a*d - b*c))/(d**2*e*(m + 3)*(m + 7)) + (e*x)**(m +
1)*sqrt(c + d*x**4)*(a**2*d**2*(m + 3)*(m + 7) - b*c*(m + 1)*(4*a*d + (m + 5)*(
2*a*d - b*c)))*hyper((1/2, m/4 + 1/4), (m/4 + 5/4,), -d*x**4/c)/(c*d**2*e*sqrt(1
+ d*x**4/c)*(m + 1)*(m + 3)*(m + 7))

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Mathematica [A]  time = 0.218335, size = 164, normalized size = 0.82 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left (a^2 \left (m^2+14 m+45\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )+b (m+1) x^4 \left (2 a (m+9) \, _2F_1\left (\frac{1}{2},\frac{m+5}{4};\frac{m+9}{4};-\frac{d x^4}{c}\right )+b (m+5) x^4 \, _2F_1\left (\frac{1}{2},\frac{m+9}{4};\frac{m+13}{4};-\frac{d x^4}{c}\right )\right )\right )}{(m+1) (m+5) (m+9) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a^2*(45 + 14*m + m^2)*Hypergeometric2F1[1/2, (1
+ m)/4, (5 + m)/4, -((d*x^4)/c)] + b*(1 + m)*x^4*(2*a*(9 + m)*Hypergeometric2F1[
1/2, (5 + m)/4, (9 + m)/4, -((d*x^4)/c)] + b*(5 + m)*x^4*Hypergeometric2F1[1/2,
(9 + m)/4, (13 + m)/4, -((d*x^4)/c)])))/((1 + m)*(5 + m)*(9 + m)*Sqrt[c + d*x^4]
)

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( b{x}^{4}+a \right ) ^{2}{\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 133.659, size = 185, normalized size = 0.92 \[ \frac{a^{2} e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} + \frac{a b e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{5}{4} \\ \frac{m}{4} + \frac{9}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right )} + \frac{b^{2} e^{m} x^{9} x^{m} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{9}{4} \\ \frac{m}{4} + \frac{13}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*e**m*x*x**m*gamma(m/4 + 1/4)*hyper((1/2, m/4 + 1/4), (m/4 + 5/4,), d*x**4*e
xp_polar(I*pi)/c)/(4*sqrt(c)*gamma(m/4 + 5/4)) + a*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)/(2*sqrt(c)*
gamma(m/4 + 9/4)) + b**2*e**m*x**9*x**m*gamma(m/4 + 9/4)*hyper((1/2, m/4 + 9/4),
(m/4 + 13/4,), d*x**4*exp_polar(I*pi)/c)/(4*sqrt(c)*gamma(m/4 + 13/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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