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

Optimal. Leaf size=173 \[ -\frac{(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{7/3}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^4}{4 b} \]

[Out]

(d*(2*b*c - a*d)*x)/b^2 + (d^2*x^4)/(4*b) - ((b*c - a*d)^2*ArcTan[(a^(1/3) - 2*b
^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(7/3)) + ((b*c - a*d)^2*Log[a^(
1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(7/3)) - ((b*c - a*d)^2*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(7/3))

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Rubi [A]  time = 0.265867, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{7/3}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

(d*(2*b*c - a*d)*x)/b^2 + (d^2*x^4)/(4*b) - ((b*c - a*d)^2*ArcTan[(a^(1/3) - 2*b
^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(7/3)) + ((b*c - a*d)^2*Log[a^(
1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(7/3)) - ((b*c - a*d)^2*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(7/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{2} x^{4}}{4 b} - \frac{\left (a d - 2 b c\right ) \int d\, dx}{b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\left (a d - b c\right )^{2} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*x**4/(4*b) - (a*d - 2*b*c)*Integral(d, x)/b**2 + (a*d - b*c)**2*log(a**(1/3
) + b**(1/3)*x)/(3*a**(2/3)*b**(7/3)) - (a*d - b*c)**2*log(a**(2/3) - a**(1/3)*b
**(1/3)*x + b**(2/3)*x**2)/(6*a**(2/3)*b**(7/3)) - sqrt(3)*(a*d - b*c)**2*atan(s
qrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(2/3)*b**(7/3))

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Mathematica [A]  time = 0.185366, size = 167, normalized size = 0.97 \[ \frac{-2 (b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+3 a^{2/3} b^{4/3} d^2 x^4-12 a^{2/3} \sqrt [3]{b} d x (a d-2 b c)+4 (b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+4 \sqrt{3} (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{12 a^{2/3} b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-12*a^(2/3)*b^(1/3)*d*(-2*b*c + a*d)*x + 3*a^(2/3)*b^(4/3)*d^2*x^4 + 4*Sqrt[3]*
(b*c - a*d)^2*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 4*(b*c - a*d)
^2*Log[a^(1/3) + b^(1/3)*x] - 2*(b*c - a*d)^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(12*a^(2/3)*b^(7/3))

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Maple [B]  time = 0.005, size = 334, normalized size = 1.9 \[{\frac{{d}^{2}{x}^{4}}{4\,b}}-{\frac{a{d}^{2}x}{{b}^{2}}}+2\,{\frac{dxc}{b}}+{\frac{{a}^{2}{d}^{2}}{3\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,acd}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{c}^{2}}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}{d}^{2}}{6\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{acd}{3\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{c}^{2}}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{a}^{2}{d}^{2}}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,\sqrt{3}cad}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{c}^{2}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^2/(b*x^3+a),x)

[Out]

1/4*d^2*x^4/b-d^2/b^2*a*x+2*d/b*x*c+1/3/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^2*d^
2-2/3/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c*a*d+1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3)
)*c^2-1/6/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^2*d^2+1/3/b^2/(a/b
)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c*a*d-1/6/b/(a/b)^(2/3)*ln(x^2-x*(a/b)
^(1/3)+(a/b)^(2/3))*c^2+1/3/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^
(1/3)*x-1))*a^2*d^2-2/3/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3
)*x-1))*c*a*d+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*
c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.212444, size = 259, normalized size = 1.5 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 4 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 12 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (b d^{2} x^{4} + 4 \,{\left (2 \, b c d - a d^{2}\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{36 \, \left (a^{2} b\right )^{\frac{1}{3}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*sqrt(3)*(2*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log((a^2*b)^(2/3)*x^2 -
(a^2*b)^(1/3)*a*x + a^2) - 4*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log((a^2*b
)^(1/3)*x + a) - 12*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(1/3*(2*sqrt(3)*(a^2*b
)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*(b*d^2*x^4 + 4*(2*b*c*d - a*d^2)*x)*(a^2*b
)^(1/3))/((a^2*b)^(1/3)*b^2)

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Sympy [A]  time = 2.99486, size = 156, normalized size = 0.9 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{7} - a^{6} d^{6} + 6 a^{5} b c d^{5} - 15 a^{4} b^{2} c^{2} d^{4} + 20 a^{3} b^{3} c^{3} d^{3} - 15 a^{2} b^{4} c^{4} d^{2} + 6 a b^{5} c^{5} d - b^{6} c^{6}, \left ( t \mapsto t \log{\left (\frac{3 t a b^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )} \right )\right )} + \frac{d^{2} x^{4}}{4 b} - \frac{x \left (a d^{2} - 2 b c d\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a**2*b**7 - a**6*d**6 + 6*a**5*b*c*d**5 - 15*a**4*b**2*c**2*d**
4 + 20*a**3*b**3*c**3*d**3 - 15*a**2*b**4*c**4*d**2 + 6*a*b**5*c**5*d - b**6*c**
6, Lambda(_t, _t*log(3*_t*a*b**2/(a**2*d**2 - 2*a*b*c*d + b**2*c**2) + x))) + d*
*2*x**4/(4*b) - x*(a*d**2 - 2*b*c*d)/b**2

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GIAC/XCAS [A]  time = 0.218411, size = 336, normalized size = 1.94 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b c d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b c d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} - \frac{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{4}} + \frac{b^{3} d^{2} x^{4} + 8 \, b^{3} c d x - 4 \, a b^{2} d^{2} x}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*((-a*b^2)^(1/3)*b^2*c^2 - 2*(-a*b^2)^(1/3)*a*b*c*d + (-a*b^2)^(1/3)*
a^2*d^2)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^3) + 1/6*((-
a*b^2)^(1/3)*b^2*c^2 - 2*(-a*b^2)^(1/3)*a*b*c*d + (-a*b^2)^(1/3)*a^2*d^2)*ln(x^2
+ x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^3) - 1/3*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2
*d^2)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^4) + 1/4*(b^3*d^2*x^4 + 8*b^3*
c*d*x - 4*a*b^2*d^2*x)/b^4