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

Optimal. Leaf size=115 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 \sqrt{c}}-\frac{2 \sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2}-\frac{\sqrt{c+d x^3}}{3 a x^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.350573, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 \sqrt{c}}-\frac{2 \sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2}-\frac{\sqrt{c+d x^3}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.2553, size = 104, normalized size = 0.9 \[ - \frac{\sqrt{c + d x^{3}}}{3 a x^{3}} - \frac{2 \sqrt{b} \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{2}} - \frac{2 \left (\frac{a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{2} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.991399, size = 407, normalized size = 3.54 \[ \frac{\frac{6 b c d x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{5 b d x^3 \left (3 a c+4 a d x^3+b c x^3+3 b d x^6\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{a \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}}{9 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((6*b*c*d*x^6*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)])/(-4*a*c*Appell
F1[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*(3*a*c + b*c*x^3 + 4*a*d*x^3 + 3*b*d*x^6)*AppellF1[3/2, 1/2
, 1, 5/2, -(c/(d*x^3)), -(a/(b*x^3))] - 3*(a + b*x^3)*(c + d*x^3)*(2*a*d*AppellF
1[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))]))/(a*(-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*x^3*(a +
b*x^3)*Sqrt[c + d*x^3])

_______________________________________________________________________________________

Maple [C]  time = 0.016, size = 518, normalized size = 4.5 \[{\frac{1}{a} \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{d{x}^{3}+c}}-{\frac{d}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \right ) }+{\frac{{b}^{2}}{{a}^{2}} \left ({\frac{2}{3\,b}\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{3}}\sqrt{2}}{b{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) }-{\frac{b}{{a}^{2}} \left ({\frac{2}{3}\sqrt{d{x}^{3}+c}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ) \sqrt{c}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a*(-1/3*(d*x^3+c)^(1/2)/x^3-1/3*d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(1/2))+1/
a^2*b^2*(2/3*(d*x^3+c)^(1/2)/b+1/3*I/b/d^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)))-b/a^2*(2/3*(d*x^3+c)^(1/2)-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))*c^(1
/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.242917, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} c - a b d} \sqrt{c} x^{3} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}}{b x^{3} + a}\right ) -{\left (2 \, b c - a d\right )} x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 2 \, \sqrt{d x^{3} + c} a \sqrt{c}}{6 \, a^{2} \sqrt{c} x^{3}}, \frac{4 \, \sqrt{-b^{2} c + a b d} \sqrt{c} x^{3} \arctan \left (\frac{\sqrt{-b^{2} c + a b d}}{\sqrt{d x^{3} + c} b}\right ) -{\left (2 \, b c - a d\right )} x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 2 \, \sqrt{d x^{3} + c} a \sqrt{c}}{6 \, a^{2} \sqrt{c} x^{3}}, -\frac{{\left (2 \, b c - a d\right )} x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - \sqrt{b^{2} c - a b d} \sqrt{-c} x^{3} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}}{b x^{3} + a}\right ) + \sqrt{d x^{3} + c} a \sqrt{-c}}{3 \, a^{2} \sqrt{-c} x^{3}}, -\frac{{\left (2 \, b c - a d\right )} x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 2 \, \sqrt{-b^{2} c + a b d} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{-b^{2} c + a b d}}{\sqrt{d x^{3} + c} b}\right ) + \sqrt{d x^{3} + c} a \sqrt{-c}}{3 \, a^{2} \sqrt{-c} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c + d*x**3)/(x**4*(a + b*x**3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221047, size = 163, normalized size = 1.42 \[ \frac{1}{3} \, d^{2}{\left (\frac{2 \,{\left (b^{2} c - a b d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} - \frac{\sqrt{d x^{3} + c}}{a d^{2} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*d^2*(2*(b^2*c - a*b*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(
-b^2*c + a*b*d)*a^2*d^2) - (2*b*c - a*d)*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(a^2*s
qrt(-c)*d^2) - sqrt(d*x^3 + c)/(a*d^2*x^3))