3.737 \(\int \frac{x^{7/2}}{a+c x^4} \, dx\)

Optimal. Leaf size=297 \[ \frac{\sqrt [8]{-a} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}+\frac{\sqrt [8]{-a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac{2 \sqrt{x}}{c} \]

[Out]

(2*Sqrt[x])/c + ((-a)^(1/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2
*Sqrt[2]*c^(9/8)) - ((-a)^(1/8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]
)/(2*Sqrt[2]*c^(9/8)) - ((-a)^(1/8)*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*c^(
9/8)) - ((-a)^(1/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*c^(9/8)) + ((-a)^(
1/8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2
]*c^(9/8)) - ((-a)^(1/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c
^(1/4)*x])/(4*Sqrt[2]*c^(9/8))

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Rubi [A]  time = 0.574625, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ \frac{\sqrt [8]{-a} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} c^{9/8}}+\frac{\sqrt [8]{-a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} c^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac{2 \sqrt{x}}{c} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(a + c*x^4),x]

[Out]

(2*Sqrt[x])/c + ((-a)^(1/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2
*Sqrt[2]*c^(9/8)) - ((-a)^(1/8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]
)/(2*Sqrt[2]*c^(9/8)) - ((-a)^(1/8)*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*c^(
9/8)) - ((-a)^(1/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*c^(9/8)) + ((-a)^(
1/8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2
]*c^(9/8)) - ((-a)^(1/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c
^(1/4)*x])/(4*Sqrt[2]*c^(9/8))

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Rubi in Sympy [A]  time = 118.395, size = 272, normalized size = 0.92 \[ \frac{2 \sqrt{x}}{c} + \frac{\sqrt{2} \sqrt [8]{- a} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 c^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{- a} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 c^{\frac{9}{8}}} - \frac{\sqrt [8]{- a} \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 c^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{- a} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{4 c^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{- a} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{4 c^{\frac{9}{8}}} - \frac{\sqrt [8]{- a} \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 c^{\frac{9}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(c*x**4+a),x)

[Out]

2*sqrt(x)/c + sqrt(2)*(-a)**(1/8)*log(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c*
*(1/4)*x + (-a)**(1/4))/(8*c**(9/8)) - sqrt(2)*(-a)**(1/8)*log(sqrt(2)*c**(1/8)*
sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(8*c**(9/8)) - (-a)**(1/8)*atan(
c**(1/8)*sqrt(x)/(-a)**(1/8))/(2*c**(9/8)) - sqrt(2)*(-a)**(1/8)*atan(sqrt(2)*c*
*(1/8)*sqrt(x)/(-a)**(1/8) - 1)/(4*c**(9/8)) - sqrt(2)*(-a)**(1/8)*atan(sqrt(2)*
c**(1/8)*sqrt(x)/(-a)**(1/8) + 1)/(4*c**(9/8)) - (-a)**(1/8)*atanh(c**(1/8)*sqrt
(x)/(-a)**(1/8))/(2*c**(9/8))

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Mathematica [A]  time = 0.368669, size = 395, normalized size = 1.33 \[ \frac{\sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-\sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+\sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-\sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-2 \sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [8]{c} \sqrt{x}}{4 c^{9/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(a + c*x^4),x]

[Out]

(8*c^(1/8)*Sqrt[x] - 2*a^(1/8)*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[
Pi/8]]*Cos[Pi/8] - 2*a^(1/8)*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi
/8]]*Cos[Pi/8] + a^(1/8)*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*S
qrt[x]*Cos[Pi/8]] - a^(1/8)*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8
)*Sqrt[x]*Cos[Pi/8]] + 2*a^(1/8)*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/
a^(1/8)]*Sin[Pi/8] - 2*a^(1/8)*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^
(1/8)]*Sin[Pi/8] + a^(1/8)*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*S
in[Pi/8]]*Sin[Pi/8] - a^(1/8)*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x
]*Sin[Pi/8]]*Sin[Pi/8])/(4*c^(9/8))

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Maple [C]  time = 0.023, size = 39, normalized size = 0.1 \[ 2\,{\frac{\sqrt{x}}{c}}-{\frac{a}{4\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(c*x^4+a),x)

[Out]

2*x^(1/2)/c-1/4*a/c^2*sum(1/_R^7*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{7}{2}}}{c x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a),x, algorithm="maxima")

[Out]

integrate(x^(7/2)/(c*x^4 + a), x)

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Fricas [A]  time = 0.262596, size = 504, normalized size = 1.7 \[ \frac{\sqrt{2}{\left (4 \, \sqrt{2} c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}}}{\sqrt{c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} + x} + \sqrt{x}}\right ) - \sqrt{2} c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \log \left (c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + \sqrt{x}\right ) + \sqrt{2} c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \log \left (-c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + \sqrt{x}\right ) + 4 \, c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}}}{c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{x} + \sqrt{2 \, c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} + 2 \, \sqrt{2} c \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + 2 \, x}}\right ) + 4 \, c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}}}{c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} - \sqrt{2} \sqrt{x} - \sqrt{2 \, c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} - 2 \, \sqrt{2} c \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + 2 \, x}}\right ) - c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \log \left (2 \, c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} + 2 \, \sqrt{2} c \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + 2 \, x\right ) + c \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} \log \left (2 \, c^{2} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{4}} - 2 \, \sqrt{2} c \sqrt{x} \left (-\frac{a}{c^{9}}\right )^{\frac{1}{8}} + 2 \, x\right ) + 8 \, \sqrt{2} \sqrt{x}\right )}}{8 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*(4*sqrt(2)*c*(-a/c^9)^(1/8)*arctan(c*(-a/c^9)^(1/8)/(sqrt(c^2*(-a/c^
9)^(1/4) + x) + sqrt(x))) - sqrt(2)*c*(-a/c^9)^(1/8)*log(c*(-a/c^9)^(1/8) + sqrt
(x)) + sqrt(2)*c*(-a/c^9)^(1/8)*log(-c*(-a/c^9)^(1/8) + sqrt(x)) + 4*c*(-a/c^9)^
(1/8)*arctan(c*(-a/c^9)^(1/8)/(c*(-a/c^9)^(1/8) + sqrt(2)*sqrt(x) + sqrt(2*c^2*(
-a/c^9)^(1/4) + 2*sqrt(2)*c*sqrt(x)*(-a/c^9)^(1/8) + 2*x))) + 4*c*(-a/c^9)^(1/8)
*arctan(-c*(-a/c^9)^(1/8)/(c*(-a/c^9)^(1/8) - sqrt(2)*sqrt(x) - sqrt(2*c^2*(-a/c
^9)^(1/4) - 2*sqrt(2)*c*sqrt(x)*(-a/c^9)^(1/8) + 2*x))) - c*(-a/c^9)^(1/8)*log(2
*c^2*(-a/c^9)^(1/4) + 2*sqrt(2)*c*sqrt(x)*(-a/c^9)^(1/8) + 2*x) + c*(-a/c^9)^(1/
8)*log(2*c^2*(-a/c^9)^(1/4) - 2*sqrt(2)*c*sqrt(x)*(-a/c^9)^(1/8) + 2*x) + 8*sqrt
(2)*sqrt(x))/c

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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(x**(7/2)/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.281345, size = 601, normalized size = 2.02 \[ -\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, c} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, c} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, c} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, c} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, c} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, c} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, c} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, c} + \frac{2 \, \sqrt{x}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + a),x, algorithm="giac")

[Out]

-1/4*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sq
rt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/c - 1/4*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*ar
ctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8
)))/c - 1/4*sqrt(-sqrt(2) + 2)*(a/c)^(1/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8)
+ 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/c - 1/4*sqrt(-sqrt(2) + 2)*(a/c)
^(1/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(
a/c)^(1/8)))/c - 1/8*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*ln(sqrt(x)*sqrt(sqrt(2) + 2)*
(a/c)^(1/8) + x + (a/c)^(1/4))/c + 1/8*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*ln(-sqrt(x)
*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/c - 1/8*sqrt(-sqrt(2) + 2)*(a/
c)^(1/8)*ln(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/c + 1/8*sq
rt(-sqrt(2) + 2)*(a/c)^(1/8)*ln(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a
/c)^(1/4))/c + 2*sqrt(x)/c