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

Optimal. Leaf size=117 \[ -\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 c^{3/2}}-\frac{\sqrt{c+d x^3}}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.361898, antiderivative size = 117, 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 \[ -\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 c^{3/2}}-\frac{\sqrt{c+d x^3}}{3 a c x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.4273, size = 104, normalized size = 0.89 \[ - \frac{\sqrt{c + d x^{3}}}{3 a c x^{3}} + \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{2} \sqrt{a d - b c}} + \frac{2 \left (\frac{a d}{2} + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{2} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.584755, size = 409, normalized size = 3.5 \[ \frac{\frac{6 b d x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{5 b d x^3 \left (3 a c+2 a d x^3+b c x^3+3 b d x^6\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{a c \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}}{9 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((6*b*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*(3*a*c + b*c*x^3 + 2*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*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))]))/(a*c*(-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])

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Maple [C]  time = 0.014, size = 498, normalized size = 4.3 \[{\frac{1}{a} \left ( -{\frac{1}{3\,c{x}^{3}}\sqrt{d{x}^{3}+c}}+{\frac{d}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) }-{\frac{{\frac{i}{3}}{b}^{2}\sqrt{2}}{{a}^{2}{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{ad-bc}\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}}+{\frac{2\,b}{3\,{a}^{2}}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256909, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, b c^{\frac{3}{2}} x^{3} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{3} + a}\right ) +{\left (2 \, b c + a d\right )} x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 2 \, \sqrt{d x^{3} + c} a \sqrt{c}}{6 \, a^{2} c^{\frac{3}{2}} x^{3}}, -\frac{4 \, b c^{\frac{3}{2}} x^{3} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{3} + c} b}\right ) -{\left (2 \, b c + a d\right )} x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c} a \sqrt{c}}{6 \, a^{2} c^{\frac{3}{2}} x^{3}}, \frac{b \sqrt{-c} c x^{3} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{3} + a}\right ) -{\left (2 \, b c + a d\right )} x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - \sqrt{d x^{3} + c} a \sqrt{-c}}{3 \, a^{2} \sqrt{-c} c x^{3}}, -\frac{2 \, b \sqrt{-c} c x^{3} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{3} + c} b}\right ) +{\left (2 \, b c + a d\right )} x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + \sqrt{d x^{3} + c} a \sqrt{-c}}{3 \, a^{2} \sqrt{-c} c x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218076, size = 159, normalized size = 1.36 \[ \frac{1}{3} \, d^{2}{\left (\frac{2 \, b^{2} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c d^{2}} - \frac{\sqrt{d x^{3} + c}}{a c d^{2} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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