3.753 \(\int \frac{x^{15/2}}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=329 \[ \frac{9 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{7/8} c^{17/8}}-\frac{9 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{7/8} c^{17/8}}+\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{7/8} c^{17/8}}-\frac{9 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{7/8} c^{17/8}}-\frac{9 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac{9 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac{9 \sqrt{x}}{64 c^2 \left (a+c x^4\right )}-\frac{x^{9/2}}{8 c \left (a+c x^4\right )^2} \]

[Out]

-x^(9/2)/(8*c*(a + c*x^4)^2) - (9*Sqrt[x])/(64*c^2*(a + c*x^4)) + (9*ArcTan[1 -
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(7/8)*c^(17/8)) - (9*Ar
cTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(7/8)*c^(17/8)
) - (9*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(7/8)*c^(17/8)) - (9*ArcT
anh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(7/8)*c^(17/8)) + (9*Log[(-a)^(1/4)
- Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(7/8)*c^(1
7/8)) - (9*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(51
2*Sqrt[2]*(-a)^(7/8)*c^(17/8))

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Rubi [A]  time = 0.596897, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ \frac{9 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{7/8} c^{17/8}}-\frac{9 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{7/8} c^{17/8}}+\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{7/8} c^{17/8}}-\frac{9 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{7/8} c^{17/8}}-\frac{9 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac{9 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{7/8} c^{17/8}}-\frac{9 \sqrt{x}}{64 c^2 \left (a+c x^4\right )}-\frac{x^{9/2}}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(15/2)/(a + c*x^4)^3,x]

[Out]

-x^(9/2)/(8*c*(a + c*x^4)^2) - (9*Sqrt[x])/(64*c^2*(a + c*x^4)) + (9*ArcTan[1 -
(Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(7/8)*c^(17/8)) - (9*Ar
cTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a)^(7/8)*c^(17/8)
) - (9*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(7/8)*c^(17/8)) - (9*ArcT
anh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(7/8)*c^(17/8)) + (9*Log[(-a)^(1/4)
- Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(512*Sqrt[2]*(-a)^(7/8)*c^(1
7/8)) - (9*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(51
2*Sqrt[2]*(-a)^(7/8)*c^(17/8))

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Rubi in Sympy [A]  time = 127.005, size = 309, normalized size = 0.94 \[ - \frac{x^{\frac{9}{2}}}{8 c \left (a + c x^{4}\right )^{2}} - \frac{9 \sqrt{x}}{64 c^{2} \left (a + c x^{4}\right )} + \frac{9 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{17}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{9 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{17}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{9 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{17}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{512 c^{\frac{17}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{512 c^{\frac{17}{8}} \left (- a\right )^{\frac{7}{8}}} - \frac{9 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{17}{8}} \left (- a\right )^{\frac{7}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(15/2)/(c*x**4+a)**3,x)

[Out]

-x**(9/2)/(8*c*(a + c*x**4)**2) - 9*sqrt(x)/(64*c**2*(a + c*x**4)) + 9*sqrt(2)*l
og(-sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4)*x + (-a)**(1/4))/(1024*c**(1
7/8)*(-a)**(7/8)) - 9*sqrt(2)*log(sqrt(2)*c**(1/8)*sqrt(x)*(-a)**(1/8) + c**(1/4
)*x + (-a)**(1/4))/(1024*c**(17/8)*(-a)**(7/8)) - 9*atan(c**(1/8)*sqrt(x)/(-a)**
(1/8))/(256*c**(17/8)*(-a)**(7/8)) - 9*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt(x)/(-a
)**(1/8) - 1)/(512*c**(17/8)*(-a)**(7/8)) - 9*sqrt(2)*atan(sqrt(2)*c**(1/8)*sqrt
(x)/(-a)**(1/8) + 1)/(512*c**(17/8)*(-a)**(7/8)) - 9*atanh(c**(1/8)*sqrt(x)/(-a)
**(1/8))/(256*c**(17/8)*(-a)**(7/8))

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Mathematica [A]  time = 0.838436, size = 428, normalized size = 1.3 \[ \frac{-\frac{9 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{7/8}}+\frac{9 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{7/8}}-\frac{9 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{7/8}}+\frac{9 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{a^{7/8}}+\frac{18 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{a^{7/8}}+\frac{18 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{a^{7/8}}-\frac{18 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{a^{7/8}}+\frac{18 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{a^{7/8}}+\frac{64 a \sqrt [8]{c} \sqrt{x}}{\left (a+c x^4\right )^2}-\frac{136 \sqrt [8]{c} \sqrt{x}}{a+c x^4}}{512 c^{17/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(15/2)/(a + c*x^4)^3,x]

[Out]

((64*a*c^(1/8)*Sqrt[x])/(a + c*x^4)^2 - (136*c^(1/8)*Sqrt[x])/(a + c*x^4) + (18*
ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8])/a^(7/8) + (18
*ArcTan[(c^(1/8)*Sqrt[x]*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8])/a^(7/8) - (9
*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]])/a^(7/
8) + (9*Cos[Pi/8]*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Cos[Pi/8]]
)/a^(7/8) - (18*ArcTan[Cot[Pi/8] - (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8
])/a^(7/8) + (18*ArcTan[Cot[Pi/8] + (c^(1/8)*Sqrt[x]*Csc[Pi/8])/a^(1/8)]*Sin[Pi/
8])/a^(7/8) - (9*Log[a^(1/4) + c^(1/4)*x - 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[Pi/8]]*
Sin[Pi/8])/a^(7/8) + (9*Log[a^(1/4) + c^(1/4)*x + 2*a^(1/8)*c^(1/8)*Sqrt[x]*Sin[
Pi/8]]*Sin[Pi/8])/a^(7/8))/(512*c^(17/8))

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Maple [C]  time = 0.028, size = 59, normalized size = 0.2 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ( -{\frac{9\,a\sqrt{x}}{128\,{c}^{2}}}-{\frac{17\,{x}^{9/2}}{128\,c}} \right ) }+{\frac{9}{512\,{c}^{3}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(15/2)/(c*x^4+a)^3,x)

[Out]

2*(-9/128*a/c^2*x^(1/2)-17/128/c*x^(9/2))/(c*x^4+a)^2+9/512/c^3*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. \[ \frac{9 \, c x^{\frac{17}{2}} + a x^{\frac{9}{2}}}{64 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} - 9 \, \int \frac{x^{\frac{7}{2}}}{128 \,{\left (a c^{2} x^{4} + a^{2} c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

1/64*(9*c*x^(17/2) + a*x^(9/2))/(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c) - 9*integrat
e(1/128*x^(7/2)/(a*c^2*x^4 + a^2*c), x)

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Fricas [A]  time = 0.266906, size = 895, normalized size = 2.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

-1/1024*sqrt(2)*(36*sqrt(2)*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^7*c^17))^(1
/8)*arctan(a*c^2*(-1/(a^7*c^17))^(1/8)/(sqrt(a^2*c^4*(-1/(a^7*c^17))^(1/4) + x)
+ sqrt(x))) - 9*sqrt(2)*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^7*c^17))^(1/8)*
log(a*c^2*(-1/(a^7*c^17))^(1/8) + sqrt(x)) + 9*sqrt(2)*(c^4*x^8 + 2*a*c^3*x^4 +
a^2*c^2)*(-1/(a^7*c^17))^(1/8)*log(-a*c^2*(-1/(a^7*c^17))^(1/8) + sqrt(x)) + 36*
(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^7*c^17))^(1/8)*arctan(a*c^2*(-1/(a^7*c^
17))^(1/8)/(a*c^2*(-1/(a^7*c^17))^(1/8) + sqrt(2)*sqrt(x) + sqrt(2*a^2*c^4*(-1/(
a^7*c^17))^(1/4) + 2*sqrt(2)*a*c^2*sqrt(x)*(-1/(a^7*c^17))^(1/8) + 2*x))) + 36*(
c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^7*c^17))^(1/8)*arctan(-a*c^2*(-1/(a^7*c^
17))^(1/8)/(a*c^2*(-1/(a^7*c^17))^(1/8) - sqrt(2)*sqrt(x) - sqrt(2*a^2*c^4*(-1/(
a^7*c^17))^(1/4) - 2*sqrt(2)*a*c^2*sqrt(x)*(-1/(a^7*c^17))^(1/8) + 2*x))) - 9*(c
^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^7*c^17))^(1/8)*log(2*a^2*c^4*(-1/(a^7*c^1
7))^(1/4) + 2*sqrt(2)*a*c^2*sqrt(x)*(-1/(a^7*c^17))^(1/8) + 2*x) + 9*(c^4*x^8 +
2*a*c^3*x^4 + a^2*c^2)*(-1/(a^7*c^17))^(1/8)*log(2*a^2*c^4*(-1/(a^7*c^17))^(1/4)
- 2*sqrt(2)*a*c^2*sqrt(x)*(-1/(a^7*c^17))^(1/8) + 2*x) + 8*sqrt(2)*(17*c*x^4 +
9*a)*sqrt(x))/(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)

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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**(15/2)/(c*x**4+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.330876, size = 659, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

9/512*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*s
qrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a*c^2) + 9/512*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)))/(a*c^2) + 9/512*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)))/(a*c^2) + 9/512
*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)))/(a*c^2) + 9/1024*sqrt(sqrt(2) + 2)*(a/c)^(
1/8)*ln(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c^2) - 9/102
4*sqrt(sqrt(2) + 2)*(a/c)^(1/8)*ln(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x +
(a/c)^(1/4))/(a*c^2) + 9/1024*sqrt(-sqrt(2) + 2)*(a/c)^(1/8)*ln(sqrt(x)*sqrt(-sq
rt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c^2) - 9/1024*sqrt(-sqrt(2) + 2)*(a
/c)^(1/8)*ln(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*c^2)
- 1/64*(17*c*x^(9/2) + 9*a*sqrt(x))/((c*x^4 + a)^2*c^2)