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

Optimal. Leaf size=198 \[ -\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (2 b^2 c^2 (m+1)-(m+3) \left (2 a^2 d^2-(m+1) (b c-a d)^2\right )\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{2 c d^2 e (m+1) (m+3) \sqrt{c+d x^4}}+\frac{(e x)^{m+1} (b c-a d)^2}{2 c d^2 e \sqrt{c+d x^4}}+\frac{b^2 \sqrt{c+d x^4} (e x)^{m+1}}{d^2 e (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.477012, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (2 b^2 c^2 (m+1)-(m+3) \left (2 a^2 d^2-(m+1) (b c-a d)^2\right )\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{2 c d^2 e (m+1) (m+3) \sqrt{c+d x^4}}+\frac{(e x)^{m+1} (b c-a d)^2}{2 c d^2 e \sqrt{c+d x^4}}+\frac{b^2 \sqrt{c+d x^4} (e x)^{m+1}}{d^2 e (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.8435, size = 165, normalized size = 0.83 \[ \frac{b^{2} \left (e x\right )^{m + 1} \sqrt{c + d x^{4}}}{d^{2} e \left (m + 3\right )} + \frac{\left (e x\right )^{m + 1} \left (a d - b c\right )^{2}}{2 c d^{2} e \sqrt{c + d x^{4}}} + \frac{\left (e x\right )^{m + 1} \sqrt{c + d x^{4}} \left (- 2 b^{2} c^{2} \left (m + 1\right ) + \left (m + 3\right ) \left (2 a^{2} d^{2} - \left (m + 1\right ) \left (a d - b c\right )^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{2 c^{2} d^{2} e \sqrt{1 + \frac{d x^{4}}{c}} \left (m + 1\right ) \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.243799, size = 167, normalized size = 0.84 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left (a^2 \left (m^2+14 m+45\right ) \, _2F_1\left (\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )+b (m+1) x^4 \left (2 a (m+9) \, _2F_1\left (\frac{3}{2},\frac{m+5}{4};\frac{m+9}{4};-\frac{d x^4}{c}\right )+b (m+5) x^4 \, _2F_1\left (\frac{3}{2},\frac{m+9}{4};\frac{m+13}{4};-\frac{d x^4}{c}\right )\right )\right )}{c (m+1) (m+5) (m+9) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a^2*(45 + 14*m + m^2)*Hypergeometric2F1[3/2, (1
+ m)/4, (5 + m)/4, -((d*x^4)/c)] + b*(1 + m)*x^4*(2*a*(9 + m)*Hypergeometric2F1[
3/2, (5 + m)/4, (9 + m)/4, -((d*x^4)/c)] + b*(5 + m)*x^4*Hypergeometric2F1[3/2,
(9 + m)/4, (13 + m)/4, -((d*x^4)/c)])))/(c*(1 + m)*(5 + m)*(9 + m)*Sqrt[c + d*x^
4])

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( b{x}^{4}+a \right ) ^{2} \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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