3.395 \(\int \frac{1}{x^4 \left (a c+b c x^3+d \sqrt{a+b x^3}\right )} \, dx\)

Optimal. Leaf size=154 \[ -\frac{b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2} \left (a c^2-d^2\right )^2}-\frac{a c-d \sqrt{a+b x^3}}{3 a x^3 \left (a c^2-d^2\right )}+\frac{2 b c^3 \log \left (c \sqrt{a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )^2}-\frac{b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]

[Out]

-(a*c - d*Sqrt[a + b*x^3])/(3*a*(a*c^2 - d^2)*x^3) - (b*d*(3*a*c^2 - d^2)*ArcTan
h[Sqrt[a + b*x^3]/Sqrt[a]])/(3*a^(3/2)*(a*c^2 - d^2)^2) - (b*c^3*Log[x])/(a*c^2
- d^2)^2 + (2*b*c^3*Log[d + c*Sqrt[a + b*x^3]])/(3*(a*c^2 - d^2)^2)

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Rubi [A]  time = 0.617852, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2} \left (a c^2-d^2\right )^2}-\frac{a c-d \sqrt{a+b x^3}}{3 a x^3 \left (a c^2-d^2\right )}+\frac{2 b c^3 \log \left (c \sqrt{a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )^2}-\frac{b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]

[Out]

-(a*c - d*Sqrt[a + b*x^3])/(3*a*(a*c^2 - d^2)*x^3) - (b*d*(3*a*c^2 - d^2)*ArcTan
h[Sqrt[a + b*x^3]/Sqrt[a]])/(3*a^(3/2)*(a*c^2 - d^2)^2) - (b*c^3*Log[x])/(a*c^2
- d^2)^2 + (2*b*c^3*Log[d + c*Sqrt[a + b*x^3]])/(3*(a*c^2 - d^2)^2)

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Rubi in Sympy [A]  time = 42.5427, size = 136, normalized size = 0.88 \[ - \frac{b c^{3} \log{\left (- b x^{3} \right )}}{3 \left (- a c^{2} + d^{2}\right )^{2}} + \frac{2 b c^{3} \log{\left (c \sqrt{a + b x^{3}} + d \right )}}{3 \left (- a c^{2} + d^{2}\right )^{2}} + \frac{a c - d \sqrt{a + b x^{3}}}{3 a x^{3} \left (- a c^{2} + d^{2}\right )} + \frac{b d \left (- 3 a c^{2} + d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}} \left (- a c^{2} + d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

-b*c**3*log(-b*x**3)/(3*(-a*c**2 + d**2)**2) + 2*b*c**3*log(c*sqrt(a + b*x**3) +
d)/(3*(-a*c**2 + d**2)**2) + (a*c - d*sqrt(a + b*x**3))/(3*a*x**3*(-a*c**2 + d*
*2)) + b*d*(-3*a*c**2 + d**2)*atanh(sqrt(a + b*x**3)/sqrt(a))/(3*a**(3/2)*(-a*c*
*2 + d**2)**2)

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Mathematica [C]  time = 6.76996, size = 860, normalized size = 5.58 \[ \frac{5 b^2 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right ) c^4}{3 \left (a c^2-d^2\right ) \sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (-5 b c^2 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right ) x^3+2 \left (a c^2-d^2\right ) F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right )+a c^2 F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right )\right )}-\frac{b \log (x) c^3}{\left (a c^2-d^2\right )^2}+\frac{b \log \left (b c^2 x^3+a c^2-d^2\right ) c^3}{3 \left (a c^2-d^2\right )^2}+\frac{2 b^2 d x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right ) c^2}{3 \sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (b \left (\left (d^2-a c^2\right ) F_1\left (2;\frac{3}{2},1;3;-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )-2 a c^2 F_1\left (2;\frac{1}{2},2;3;-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )\right ) x^3+4 a \left (a c^2-d^2\right ) F_1\left (1;\frac{1}{2},1;2;-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )\right )}-\frac{5 b^2 d^3 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right ) c^2}{9 a \left (a c^2-d^2\right ) \sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (-5 b c^2 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right ) x^3+2 \left (a c^2-d^2\right ) F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right )+a c^2 F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right )\right )}-\frac{c}{3 \left (a c^2-d^2\right ) x^3}+\frac{d \sqrt{b x^3+a}}{3 a \left (a c^2-d^2\right ) x^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^4*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]

[Out]

-c/(3*(a*c^2 - d^2)*x^3) + (d*Sqrt[a + b*x^3])/(3*a*(a*c^2 - d^2)*x^3) + (2*b^2*
c^2*d*x^3*AppellF1[1, 1/2, 1, 2, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))])/(3
*Sqrt[a + b*x^3]*(a*c^2 - d^2 + b*c^2*x^3)*(4*a*(a*c^2 - d^2)*AppellF1[1, 1/2, 1
, 2, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] + b*x^3*(-2*a*c^2*AppellF1[2, 1
/2, 2, 3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] + (-(a*c^2) + d^2)*AppellF
1[2, 3/2, 1, 3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))]))) + (5*b^2*c^4*d*x^
3*AppellF1[3/2, 1/2, 1, 5/2, -(a/(b*x^3)), (-(a*c^2) + d^2)/(b*c^2*x^3)])/(3*(a*
c^2 - d^2)*Sqrt[a + b*x^3]*(a*c^2 - d^2 + b*c^2*x^3)*(-5*b*c^2*x^3*AppellF1[3/2,
1/2, 1, 5/2, -(a/(b*x^3)), (-(a*c^2) + d^2)/(b*c^2*x^3)] + 2*(a*c^2 - d^2)*Appe
llF1[5/2, 1/2, 2, 7/2, -(a/(b*x^3)), (-(a*c^2) + d^2)/(b*c^2*x^3)] + a*c^2*Appel
lF1[5/2, 3/2, 1, 7/2, -(a/(b*x^3)), (-(a*c^2) + d^2)/(b*c^2*x^3)])) - (5*b^2*c^2
*d^3*x^3*AppellF1[3/2, 1/2, 1, 5/2, -(a/(b*x^3)), (-(a*c^2) + d^2)/(b*c^2*x^3)])
/(9*a*(a*c^2 - d^2)*Sqrt[a + b*x^3]*(a*c^2 - d^2 + b*c^2*x^3)*(-5*b*c^2*x^3*Appe
llF1[3/2, 1/2, 1, 5/2, -(a/(b*x^3)), (-(a*c^2) + d^2)/(b*c^2*x^3)] + 2*(a*c^2 -
d^2)*AppellF1[5/2, 1/2, 2, 7/2, -(a/(b*x^3)), (-(a*c^2) + d^2)/(b*c^2*x^3)] + a*
c^2*AppellF1[5/2, 3/2, 1, 7/2, -(a/(b*x^3)), (-(a*c^2) + d^2)/(b*c^2*x^3)])) - (
b*c^3*Log[x])/(a*c^2 - d^2)^2 + (b*c^3*Log[a*c^2 - d^2 + b*c^2*x^3])/(3*(a*c^2 -
d^2)^2)

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Maple [C]  time = 0.053, size = 863, normalized size = 5.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)

[Out]

-1/3*c/(a*c^2-d^2)/x^3-2*b*c^3*ln(x)/(a*c^2-d^2)^2+1/a*c*b/(a*c^2-d^2)^2*ln(x)*d
^2+1/3*a*c^5*b/(a*c^2-d^2)^2/d^2*ln(b*c^2*x^3+a*c^2-d^2)+b*c/a/(a*c^2-d^2)*ln(x)
-1/3*b*c^3/(a*c^2-d^2)/d^2*ln(b*c^2*x^3+a*c^2-d^2)+1/3*d/a/(a*c^2-d^2)*(b*x^3+a)
^(1/2)/x^3+1/3*d/a^(3/2)/(a*c^2-d^2)*b*arctanh((b*x^3+a)^(1/2)/a^(1/2))-2/3*b*c^
4/(a*c^2-d^2)^2/d*(b*x^3+a)^(1/2)-1/3*I/b*c^4/(a*c^2-d^2)^2/d*2^(1/2)*sum((-a*b^
2)^(1/3)*(1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(
1/3))^(1/2)*(b*(x-1/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3
)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)
^(1/3))^(1/2)/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*b^2)^(2/3
)*3^(1/2)+2*_alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticPi(1/3*
3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a
*b^2)^(1/3))^(1/2),-1/2*c^2/b*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*3^(1/2)*(
-a*b^2)^(2/3)*_alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)
/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))
,_alpha=RootOf(_Z^3*b*c^2+a*c^2-d^2))+2/3*b/a^2/d*(b*x^3+a)^(1/2)+4/3*d*b/a/(a*c
^2-d^2)^2*(b*x^3+a)^(1/2)*c^2-2/3*b/a^2/(a*c^2-d^2)^2*(b*x^3+a)^(1/2)*d^3-4/3*d*
b/a^(1/2)/(a*c^2-d^2)^2*arctanh((b*x^3+a)^(1/2)/a^(1/2))*c^2+2/3*b/a^(3/2)/(a*c^
2-d^2)^2*arctanh((b*x^3+a)^(1/2)/a^(1/2))*d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b c x^{3} + a c + \sqrt{b x^{3} + a} d\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^4), x)

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Fricas [A]  time = 0.394049, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a^{\frac{3}{2}} b c^{3} x^{3} \log \left (\sqrt{b x^{3} + a} c + d\right ) - 2 \, a^{\frac{3}{2}} b c^{3} x^{3} \log \left (\sqrt{b x^{3} + a} c - d\right ) -{\left (3 \, a b c^{2} d - b d^{3}\right )} x^{3} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{3} + a} \sqrt{a} + 2 \,{\left (a b c^{3} x^{3} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - 3 \, a b c^{3} x^{3} \log \left (x\right ) - a^{2} c^{3} + a c d^{2}\right )} \sqrt{a}}{6 \,{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{a} x^{3}}, \frac{\sqrt{-a} a b c^{3} x^{3} \log \left (\sqrt{b x^{3} + a} c + d\right ) - \sqrt{-a} a b c^{3} x^{3} \log \left (\sqrt{b x^{3} + a} c - d\right ) +{\left (3 \, a b c^{2} d - b d^{3}\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{3} + a} \sqrt{-a} +{\left (a b c^{3} x^{3} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - 3 \, a b c^{3} x^{3} \log \left (x\right ) - a^{2} c^{3} + a c d^{2}\right )} \sqrt{-a}}{3 \,{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{-a} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^4),x, algorithm="fricas")

[Out]

[1/6*(2*a^(3/2)*b*c^3*x^3*log(sqrt(b*x^3 + a)*c + d) - 2*a^(3/2)*b*c^3*x^3*log(s
qrt(b*x^3 + a)*c - d) - (3*a*b*c^2*d - b*d^3)*x^3*log(((b*x^3 + 2*a)*sqrt(a) + 2
*sqrt(b*x^3 + a)*a)/x^3) + 2*(a*c^2*d - d^3)*sqrt(b*x^3 + a)*sqrt(a) + 2*(a*b*c^
3*x^3*log(b*c^2*x^3 + a*c^2 - d^2) - 3*a*b*c^3*x^3*log(x) - a^2*c^3 + a*c*d^2)*s
qrt(a))/((a^3*c^4 - 2*a^2*c^2*d^2 + a*d^4)*sqrt(a)*x^3), 1/3*(sqrt(-a)*a*b*c^3*x
^3*log(sqrt(b*x^3 + a)*c + d) - sqrt(-a)*a*b*c^3*x^3*log(sqrt(b*x^3 + a)*c - d)
+ (3*a*b*c^2*d - b*d^3)*x^3*arctan(a/(sqrt(b*x^3 + a)*sqrt(-a))) + (a*c^2*d - d^
3)*sqrt(b*x^3 + a)*sqrt(-a) + (a*b*c^3*x^3*log(b*c^2*x^3 + a*c^2 - d^2) - 3*a*b*
c^3*x^3*log(x) - a^2*c^3 + a*c*d^2)*sqrt(-a))/((a^3*c^4 - 2*a^2*c^2*d^2 + a*d^4)
*sqrt(-a)*x^3)]

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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(1/x**4/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.289417, size = 275, normalized size = 1.79 \[ \frac{1}{3} \,{\left (\frac{2 \, c^{4}{\rm ln}\left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{a^{2} c^{5} - 2 \, a c^{3} d^{2} + c d^{4}} - \frac{c^{3}{\rm ln}\left (b x^{3}\right )}{a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}} + \frac{{\left (3 \, a c^{2} d - d^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{-a}} - \frac{a^{2} c^{3} - a c d^{2} -{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{3} + a}}{{\left (a c^{2} - d^{2}\right )}^{2} a b x^{3}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^4),x, algorithm="giac")

[Out]

1/3*(2*c^4*ln(abs(sqrt(b*x^3 + a)*c + d))/(a^2*c^5 - 2*a*c^3*d^2 + c*d^4) - c^3*
ln(b*x^3)/(a^2*c^4 - 2*a*c^2*d^2 + d^4) + (3*a*c^2*d - d^3)*arctan(sqrt(b*x^3 +
a)/sqrt(-a))/((a^3*c^4 - 2*a^2*c^2*d^2 + a*d^4)*sqrt(-a)) - (a^2*c^3 - a*c*d^2 -
(a*c^2*d - d^3)*sqrt(b*x^3 + a))/((a*c^2 - d^2)^2*a*b*x^3))*b