3.494 \(\int \frac{1}{x^4 \left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=241 \[ -\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^3 c^{5/2}}-\frac{d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{3 a^2 c^2 \sqrt{c+d x^3} (b c-a d)^2}-\frac{b (2 b c-a d)}{3 a^2 c \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}-\frac{1}{3 a c x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

[Out]

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

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Rubi [A]  time = 1.16483, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^3 c^{5/2}}-\frac{d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{3 a^2 c^2 \sqrt{c+d x^3} (b c-a d)^2}-\frac{b (2 b c-a d)}{3 a^2 c \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}-\frac{1}{3 a c x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 117.071, size = 216, normalized size = 0.9 \[ - \frac{1}{3 a c x^{3} \left (a + b x^{3}\right ) \sqrt{c + d x^{3}}} - \frac{b \left (a d - 2 b c\right )}{3 a^{2} c \left (a + b x^{3}\right ) \sqrt{c + d x^{3}} \left (a d - b c\right )} - \frac{d \left (3 a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{3 a^{2} c^{2} \sqrt{c + d x^{3}} \left (a d - b c\right )^{2}} - \frac{b^{\frac{5}{2}} \left (7 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{3} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{\left (3 a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{3} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*a*c*x**3*(a + b*x**3)*sqrt(c + d*x**3)) - b*(a*d - 2*b*c)/(3*a**2*c*(a + b
*x**3)*sqrt(c + d*x**3)*(a*d - b*c)) - d*(3*a**2*d**2 - 2*a*b*c*d + 2*b**2*c**2)
/(3*a**2*c**2*sqrt(c + d*x**3)*(a*d - b*c)**2) - b**(5/2)*(7*a*d - 4*b*c)*atan(s
qrt(b)*sqrt(c + d*x**3)/sqrt(a*d - b*c))/(3*a**3*(a*d - b*c)**(5/2)) + (3*a*d +
4*b*c)*atanh(sqrt(c + d*x**3)/sqrt(c))/(3*a**3*c**(5/2))

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Mathematica [C]  time = 2.04353, size = 582, normalized size = 2.41 \[ \frac{\frac{6 a b c d x^6 \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right ) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}-\frac{3 \left (a^3 d^2 \left (c+3 d x^3\right )+a^2 b d \left (-2 c^2-c d x^3+3 d^2 x^6\right )+a b^2 c \left (c^2-c d x^3-2 d^2 x^6\right )+2 b^3 c^2 x^3 \left (c+d x^3\right )\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )-5 b d x^3 \left (3 a^3 d^2 \left (c+2 d x^3\right )+a^2 b d \left (-6 c^2-c d x^3+9 d^2 x^6\right )+a b^2 c \left (3 c^2+2 c d x^3-6 d^2 x^6\right )+2 b^3 c^2 x^3 \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}{-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}}{9 a^2 c^2 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

((6*a*b*c*d*(2*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*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*AppellF
1[2, 3/2, 1, 3, -((d*x^3)/c), -((b*x^3)/a)])) - (-5*b*d*x^3*(3*a^3*d^2*(c + 2*d*
x^3) + 2*b^3*c^2*x^3*(c + 3*d*x^3) + a*b^2*c*(3*c^2 + 2*c*d*x^3 - 6*d^2*x^6) + a
^2*b*d*(-6*c^2 - c*d*x^3 + 9*d^2*x^6))*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)),
-(a/(b*x^3))] + 3*(2*b^3*c^2*x^3*(c + d*x^3) + a^3*d^2*(c + 3*d*x^3) + a*b^2*c*(
c^2 - c*d*x^3 - 2*d^2*x^6) + a^2*b*d*(-2*c^2 - c*d*x^3 + 3*d^2*x^6))*(2*a*d*Appe
llF1[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*c^2*(b
*c - a*d)^2*x^3*(a + b*x^3)*Sqrt[c + d*x^3])

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Maple [C]  time = 0.018, size = 1067, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a^2*(-1/3*(d*x^3+c)^(1/2)/c^2/x^3-2/3*d/c^2/((x^3+c/d)*d)^(1/2)+d*arctanh((d*x
^3+c)^(1/2)/c^(1/2))/c^(5/2))+1/a^2*b^2*(-1/3*b/(a*d-b*c)^2*(d*x^3+c)^(1/2)/(b*x
^3+a)-2/3*d/(a*d-b*c)^2/((x^3+c/d)*d)^(1/2)+1/2*I*b/d*2^(1/2)*sum(1/(a*d-b*c)^3*
(-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))*Elliptic
Pi(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*_alph
a*(-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*(2/3/c/((x^3+c/d)*d)^(1/2)-2/3*arct
anh((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))+2/a^3*b^2*(-2/3/(a*d-b*c)/((x^3+c/d)*d)^(1
/2)-1/3*I/d^2*b*2^(1/2)*sum(1/(-a*d+b*c)/(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)*_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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{2}{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.792627, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x^4),x, algorithm="fricas")

[Out]

[-1/6*(((4*b^4*c^3 - 7*a*b^3*c^2*d)*x^6 + (4*a*b^3*c^3 - 7*a^2*b^2*c^2*d)*x^3)*s
qrt(d*x^3 + c)*sqrt(c)*sqrt(b/(b*c - a*d))*log((b*d*x^3 + 2*b*c - a*d + 2*sqrt(d
*x^3 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^3 + a)) - ((4*b^4*c^3 - 5*a*b^3*
c^2*d - 2*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*x^6 + (4*a*b^3*c^3 - 5*a^2*b^2*c^2*d - 2*
a^3*b*c*d^2 + 3*a^4*d^3)*x^3)*sqrt(d*x^3 + c)*log(((d*x^3 + 2*c)*sqrt(c) + 2*sqr
t(d*x^3 + c)*c)/x^3) + 2*(a^2*b^2*c^3 - 2*a^3*b*c^2*d + a^4*c*d^2 + (2*a*b^3*c^2
*d - 2*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*x^6 + (2*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c
*d^2 + 3*a^4*d^3)*x^3)*sqrt(c))/(((a^3*b^3*c^4 - 2*a^4*b^2*c^3*d + a^5*b*c^2*d^2
)*x^6 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d + a^6*c^2*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt(c)
), -1/6*(2*((4*b^4*c^3 - 7*a*b^3*c^2*d)*x^6 + (4*a*b^3*c^3 - 7*a^2*b^2*c^2*d)*x^
3)*sqrt(d*x^3 + c)*sqrt(c)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c
- a*d))/(sqrt(d*x^3 + c)*b)) - ((4*b^4*c^3 - 5*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 +
3*a^3*b*d^3)*x^6 + (4*a*b^3*c^3 - 5*a^2*b^2*c^2*d - 2*a^3*b*c*d^2 + 3*a^4*d^3)*x
^3)*sqrt(d*x^3 + c)*log(((d*x^3 + 2*c)*sqrt(c) + 2*sqrt(d*x^3 + c)*c)/x^3) + 2*(
a^2*b^2*c^3 - 2*a^3*b*c^2*d + a^4*c*d^2 + (2*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + 3*a
^3*b*d^3)*x^6 + (2*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 + 3*a^4*d^3)*x^3)*sqr
t(c))/(((a^3*b^3*c^4 - 2*a^4*b^2*c^3*d + a^5*b*c^2*d^2)*x^6 + (a^4*b^2*c^4 - 2*a
^5*b*c^3*d + a^6*c^2*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt(c)), -1/6*(((4*b^4*c^3 - 7*a
*b^3*c^2*d)*x^6 + (4*a*b^3*c^3 - 7*a^2*b^2*c^2*d)*x^3)*sqrt(d*x^3 + c)*sqrt(-c)*
sqrt(b/(b*c - a*d))*log((b*d*x^3 + 2*b*c - a*d + 2*sqrt(d*x^3 + c)*(b*c - a*d)*s
qrt(b/(b*c - a*d)))/(b*x^3 + a)) + 2*((4*b^4*c^3 - 5*a*b^3*c^2*d - 2*a^2*b^2*c*d
^2 + 3*a^3*b*d^3)*x^6 + (4*a*b^3*c^3 - 5*a^2*b^2*c^2*d - 2*a^3*b*c*d^2 + 3*a^4*d
^3)*x^3)*sqrt(d*x^3 + c)*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) + 2*(a^2*b^2*c^3 -
2*a^3*b*c^2*d + a^4*c*d^2 + (2*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*x^6
+ (2*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 + 3*a^4*d^3)*x^3)*sqrt(-c))/(((a^3
*b^3*c^4 - 2*a^4*b^2*c^3*d + a^5*b*c^2*d^2)*x^6 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d +
a^6*c^2*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt(-c)), -1/3*(((4*b^4*c^3 - 7*a*b^3*c^2*d)
*x^6 + (4*a*b^3*c^3 - 7*a^2*b^2*c^2*d)*x^3)*sqrt(d*x^3 + c)*sqrt(-c)*sqrt(-b/(b*
c - a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x^3 + c)*b)) + ((4*b^
4*c^3 - 5*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*x^6 + (4*a*b^3*c^3 - 5*a^
2*b^2*c^2*d - 2*a^3*b*c*d^2 + 3*a^4*d^3)*x^3)*sqrt(d*x^3 + c)*arctan(c/(sqrt(d*x
^3 + c)*sqrt(-c))) + (a^2*b^2*c^3 - 2*a^3*b*c^2*d + a^4*c*d^2 + (2*a*b^3*c^2*d -
2*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*x^6 + (2*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2
+ 3*a^4*d^3)*x^3)*sqrt(-c))/(((a^3*b^3*c^4 - 2*a^4*b^2*c^3*d + a^5*b*c^2*d^2)*x
^6 + (a^4*b^2*c^4 - 2*a^5*b*c^3*d + a^6*c^2*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt(-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(1/x**4/(b*x**3+a)**2/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233655, size = 510, normalized size = 2.12 \[ \frac{1}{3} \, d^{3}{\left (\frac{{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x^{3} + c\right )}^{2} b^{3} c^{2} - 2 \,{\left (d x^{3} + c\right )} b^{3} c^{3} - 2 \,{\left (d x^{3} + c\right )}^{2} a b^{2} c d + 3 \,{\left (d x^{3} + c\right )} a b^{2} c^{2} d + 3 \,{\left (d x^{3} + c\right )}^{2} a^{2} b d^{2} - 7 \,{\left (d x^{3} + c\right )} a^{2} b c d^{2} + 2 \, a^{2} b c^{2} d^{2} + 3 \,{\left (d x^{3} + c\right )} a^{3} d^{3} - 2 \, a^{3} c d^{3}}{{\left (a^{2} b^{2} c^{4} d^{2} - 2 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )}{\left ({\left (d x^{3} + c\right )}^{\frac{5}{2}} b - 2 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} b c + \sqrt{d x^{3} + c} b c^{2} +{\left (d x^{3} + c\right )}^{\frac{3}{2}} a d - \sqrt{d x^{3} + c} a c d\right )}} - \frac{{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c^{2} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x^4),x, algorithm="giac")

[Out]

1/3*d^3*((4*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/((
a^3*b^2*c^2*d^3 - 2*a^4*b*c*d^4 + a^5*d^5)*sqrt(-b^2*c + a*b*d)) - (2*(d*x^3 + c
)^2*b^3*c^2 - 2*(d*x^3 + c)*b^3*c^3 - 2*(d*x^3 + c)^2*a*b^2*c*d + 3*(d*x^3 + c)*
a*b^2*c^2*d + 3*(d*x^3 + c)^2*a^2*b*d^2 - 7*(d*x^3 + c)*a^2*b*c*d^2 + 2*a^2*b*c^
2*d^2 + 3*(d*x^3 + c)*a^3*d^3 - 2*a^3*c*d^3)/((a^2*b^2*c^4*d^2 - 2*a^3*b*c^3*d^3
+ a^4*c^2*d^4)*((d*x^3 + c)^(5/2)*b - 2*(d*x^3 + c)^(3/2)*b*c + sqrt(d*x^3 + c)
*b*c^2 + (d*x^3 + c)^(3/2)*a*d - sqrt(d*x^3 + c)*a*c*d)) - (4*b*c + 3*a*d)*arcta
n(sqrt(d*x^3 + c)/sqrt(-c))/(a^3*sqrt(-c)*c^2*d^3))