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

Optimal. Leaf size=104 \[ \frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a}+\frac{2 d \sqrt{c+d x^3}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.332113, antiderivative size = 104, 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 \[ \frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a}+\frac{2 d \sqrt{c+d x^3}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.6876, size = 90, normalized size = 0.87 \[ \frac{2 d \sqrt{c + d x^{3}}}{3 b} - \frac{2 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a} - \frac{2 \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.535049, size = 325, normalized size = 3.12 \[ \frac{2 d \left (\frac{5 b^2 c^2 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}{\left (a+b x^3\right ) \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}+\frac{6 a c x^3 (a d-2 b c) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{\left (a+b x^3\right ) \left (x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}+3 \left (c+d x^3\right )\right )}{9 b \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.014, size = 565, normalized size = 5.4 \[{\frac{1}{a} \left ({\frac{2\,d{x}^{3}}{9}\sqrt{d{x}^{3}+c}}+{\frac{8\,c}{9}\sqrt{d{x}^{3}+c}}-{\frac{2}{3}{c}^{{\frac{3}{2}}}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ) } \right ) }-{\frac{b}{a} \left ({\frac{2\,d{x}^{3}}{9\,b}\sqrt{d{x}^{3}+c}}+{\frac{2}{3\,d} \left ( -{\frac{d \left ( ad-2\,bc \right ) }{{b}^{2}}}-{\frac{2\,cd}{3\,b}} \right ) \sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{3}}\sqrt{2}}{{b}^{2}{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{-{a}^{2}{d}^{2}+2\,cabd-{b}^{2}{c}^{2}}{ad-bc}\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a*(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)))-b/a*(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)*_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. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (b x^{3} + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.275508, size = 1, normalized size = 0.01 \[ \left [\frac{b c^{\frac{3}{2}} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c} a d -{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right )}{3 \, a b}, \frac{b c^{\frac{3}{2}} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c} a d + 2 \,{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right )}{3 \, a b}, -\frac{2 \, b \sqrt{-c} c \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) - 2 \, \sqrt{d x^{3} + c} a d +{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right )}{3 \, a b}, -\frac{2 \,{\left (b \sqrt{-c} c \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) - \sqrt{d x^{3} + c} a d -{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right )\right )}}{3 \, a b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220977, size = 159, normalized size = 1.53 \[ \frac{2}{3} \, d{\left (\frac{c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} d} + \frac{\sqrt{d x^{3} + c}}{b} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a b d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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