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

Optimal. Leaf size=170 \[ \frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^3 \sqrt{b}}-\frac{\sqrt{c+d x^3} (2 b c-a d)}{3 a^2 \left (a+b x^3\right )}-\frac{c \sqrt{c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \]

[Out]

-((2*b*c - a*d)*Sqrt[c + d*x^3])/(3*a^2*(a + b*x^3)) - (c*Sqrt[c + d*x^3])/(3*a*
x^3*(a + b*x^3)) + (Sqrt[c]*(4*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3
*a^3) - (Sqrt[b*c - a*d]*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*
c - a*d]])/(3*a^3*Sqrt[b])

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Rubi [A]  time = 0.748304, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^3 \sqrt{b}}-\frac{\sqrt{c+d x^3} (2 b c-a d)}{3 a^2 \left (a+b x^3\right )}-\frac{c \sqrt{c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((2*b*c - a*d)*Sqrt[c + d*x^3])/(3*a^2*(a + b*x^3)) - (c*Sqrt[c + d*x^3])/(3*a*
x^3*(a + b*x^3)) + (Sqrt[c]*(4*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3
*a^3) - (Sqrt[b*c - a*d]*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*
c - a*d]])/(3*a^3*Sqrt[b])

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Rubi in Sympy [A]  time = 79.1586, size = 153, normalized size = 0.9 \[ - \frac{\sqrt{c + d x^{3}} \left (a d - b c\right )}{3 a b x^{3} \left (a + b x^{3}\right )} + \frac{\sqrt{c + d x^{3}} \left (a d - 2 b c\right )}{3 a^{2} b x^{3}} - \frac{\sqrt{c} \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{3}} + \frac{\left (a d - 4 b c\right ) \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{3} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)*(a*d - b*c)/(3*a*b*x**3*(a + b*x**3)) + sqrt(c + d*x**3)*(a*d
- 2*b*c)/(3*a**2*b*x**3) - sqrt(c)*(3*a*d - 4*b*c)*atanh(sqrt(c + d*x**3)/sqrt(c
))/(3*a**3) + (a*d - 4*b*c)*sqrt(a*d - b*c)*atan(sqrt(b)*sqrt(c + d*x**3)/sqrt(a
*d - b*c))/(3*a**3*sqrt(b))

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Mathematica [C]  time = 0.808406, size = 439, normalized size = 2.58 \[ \frac{\frac{5 b d x^3 \left (3 a \left (c^2+c d x^3-d^2 x^6\right )+2 b c x^3 \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (c+d x^3\right ) \left (a \left (c-d x^3\right )+2 b c x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}+\frac{6 a c d x^6 (a d-2 b c) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )-x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}}{9 a^2 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((6*a*c*d*(-2*b*c + a*d)*x^6*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)])
/(4*a*c*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)] - x^3*(2*b*c*AppellF1
[2, 1/2, 2, 3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[2, 3/2, 1, 3, -((d*x^3
)/c), -((b*x^3)/a)])) + (5*b*d*x^3*(2*b*c*x^3*(c + 3*d*x^3) + 3*a*(c^2 + c*d*x^3
- d^2*x^6))*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), -(a/(b*x^3))] - 3*(c + d*x
^3)*(2*b*c*x^3 + a*(c - d*x^3))*(2*a*d*AppellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^3)),
-(a/(b*x^3))] + b*c*AppellF1[5/2, 3/2, 1, 7/2, -(c/(d*x^3)), -(a/(b*x^3))]))/(-5
*b*d*x^3*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), -(a/(b*x^3))] + 2*a*d*AppellF1
[5/2, 1/2, 2, 7/2, -(c/(d*x^3)), -(a/(b*x^3))] + b*c*AppellF1[5/2, 3/2, 1, 7/2,
-(c/(d*x^3)), -(a/(b*x^3))]))/(9*a^2*x^3*(a + b*x^3)*Sqrt[c + d*x^3])

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Maple [C]  time = 0.02, size = 1093, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a^2*(-1/3*c*(d*x^3+c)^(1/2)/x^3+2/3*d*(d*x^3+c)^(1/2)-c^(1/2)*d*arctanh((d*x^3
+c)^(1/2)/c^(1/2)))+1/a^2*b^2*(1/3*(a*d-b*c)/b^2*(d*x^3+c)^(1/2)/(b*x^3+a)+2/3*d
*(d*x^3+c)^(1/2)/b^2+1/2*I/d/b^2*2^(1/2)*sum((-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-
I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^
2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d
*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2
)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c
*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)
^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(
2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*
c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d
*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b+a))
)-2*b/a^3*(2/9*d*x^3*(d*x^3+c)^(1/2)+8/9*c*(d*x^3+c)^(1/2)-2/3*c^(3/2)*arctanh((
d*x^3+c)^(1/2)/c^(1/2)))+2/a^3*b^2*(2/9/b*d*x^3*(d*x^3+c)^(1/2)+2/3*(-d*(a*d-2*b
*c)/b^2-2/3/b*d*c)/d*(d*x^3+c)^(1/2)+1/3*I/b^2/d^2*2^(1/2)*sum((-a^2*d^2+2*a*b*c
*d-b^2*c^2)/(a*d-b*c)*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3
)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^
(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1
/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_al
pha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-
c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*
(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*_alpha^2*(-c*d^2)^(
1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(
2/3)-3*c*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3
^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b+a)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255062, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(((4*b^2*c - a*b*d)*x^6 + (4*a*b*c - a^2*d)*x^3)*sqrt((b*c - a*d)/b)*log((
b*d*x^3 + 2*b*c - a*d + 2*sqrt(d*x^3 + c)*b*sqrt((b*c - a*d)/b))/(b*x^3 + a)) +
((4*b^2*c - 3*a*b*d)*x^6 + (4*a*b*c - 3*a^2*d)*x^3)*sqrt(c)*log((d*x^3 - 2*sqrt(
d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 2*((2*a*b*c - a^2*d)*x^3 + a^2*c)*sqrt(d*x^3 +
c))/(a^3*b*x^6 + a^4*x^3), -1/6*(2*((4*b^2*c - a*b*d)*x^6 + (4*a*b*c - a^2*d)*x^
3)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x^3 + c)/sqrt(-(b*c - a*d)/b)) + ((4*b^2*c
- 3*a*b*d)*x^6 + (4*a*b*c - 3*a^2*d)*x^3)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c
)*sqrt(c) + 2*c)/x^3) + 2*((2*a*b*c - a^2*d)*x^3 + a^2*c)*sqrt(d*x^3 + c))/(a^3*
b*x^6 + a^4*x^3), 1/6*(2*((4*b^2*c - 3*a*b*d)*x^6 + (4*a*b*c - 3*a^2*d)*x^3)*sqr
t(-c)*arctan(sqrt(d*x^3 + c)/sqrt(-c)) - ((4*b^2*c - a*b*d)*x^6 + (4*a*b*c - a^2
*d)*x^3)*sqrt((b*c - a*d)/b)*log((b*d*x^3 + 2*b*c - a*d + 2*sqrt(d*x^3 + c)*b*sq
rt((b*c - a*d)/b))/(b*x^3 + a)) - 2*((2*a*b*c - a^2*d)*x^3 + a^2*c)*sqrt(d*x^3 +
c))/(a^3*b*x^6 + a^4*x^3), 1/3*(((4*b^2*c - 3*a*b*d)*x^6 + (4*a*b*c - 3*a^2*d)*
x^3)*sqrt(-c)*arctan(sqrt(d*x^3 + c)/sqrt(-c)) - ((4*b^2*c - a*b*d)*x^6 + (4*a*b
*c - a^2*d)*x^3)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x^3 + c)/sqrt(-(b*c - a*d)/b
)) - ((2*a*b*c - a^2*d)*x^3 + a^2*c)*sqrt(d*x^3 + c))/(a^3*b*x^6 + a^4*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((d*x**3+c)**(3/2)/x**4/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225684, size = 300, normalized size = 1.76 \[ -\frac{1}{3} \, d^{3}{\left (\frac{2 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} b c - 2 \, \sqrt{d x^{3} + c} b c^{2} -{\left (d x^{3} + c\right )}^{\frac{3}{2}} a d + 2 \, \sqrt{d x^{3} + c} a c d}{{\left ({\left (d x^{3} + c\right )}^{2} b - 2 \,{\left (d x^{3} + c\right )} b c + b c^{2} +{\left (d x^{3} + c\right )} a d - a c d\right )} a^{2} d^{2}} - \frac{{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3} d^{3}} + \frac{{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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