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

Optimal. Leaf size=186 \[ \frac{a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{2/3}}-\frac{a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}-\frac{a \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{2/3}}+\frac{b^2 \log \left (c+d x^3\right )}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.331619, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{2/3}}-\frac{a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}-\frac{a \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{2/3}}+\frac{b^2 \log \left (c+d x^3\right )}{3 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/(c + d*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 42.178, size = 175, normalized size = 0.94 \[ \frac{a \left (a \sqrt [3]{d} - 2 b \sqrt [3]{c}\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} d^{\frac{2}{3}}} - \frac{a \left (a \sqrt [3]{d} - 2 b \sqrt [3]{c}\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} d^{\frac{2}{3}}} - \frac{\sqrt{3} a \left (a \sqrt [3]{d} + 2 b \sqrt [3]{c}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} d^{\frac{2}{3}}} + \frac{b^{2} \log{\left (c + d x^{3} \right )}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(a*d**(1/3) - 2*b*c**(1/3))*log(c**(1/3) + d**(1/3)*x)/(3*c**(2/3)*d**(2/3)) -
a*(a*d**(1/3) - 2*b*c**(1/3))*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**
2)/(6*c**(2/3)*d**(2/3)) - sqrt(3)*a*(a*d**(1/3) + 2*b*c**(1/3))*atan(sqrt(3)*(c
**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(2/3)*d**(2/3)) + b**2*log(c + d*x**
3)/(3*d)

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Mathematica [A]  time = 0.165063, size = 200, normalized size = 1.08 \[ -\frac{\left (a^2 \sqrt [3]{c} \sqrt [3]{d}-2 a b c^{2/3}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c d^{2/3}}+\frac{\left (a^2 \sqrt [3]{c} \sqrt [3]{d}-2 a b c^{2/3}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c d^{2/3}}+\frac{\left (a^2 \sqrt [3]{c} \sqrt [3]{d}+2 a b c^{2/3}\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} x-\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c d^{2/3}}+\frac{b^2 \log \left (c+d x^3\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 211, normalized size = 1.1 \[{\frac{{a}^{2}}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{6\,d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}\sqrt{3}}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,ab}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{ab}{3\,d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{2\,\sqrt{3}ab}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{b}^{2}\ln \left ( d{x}^{3}+c \right ) }{3\,d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(d*x^3+c),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 1.98675, size = 156, normalized size = 0.84 \[ \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{3} - 27 t^{2} b^{2} c^{2} d^{2} + t \left (18 a^{3} b c d^{2} + 9 b^{4} c^{2} d\right ) - a^{6} d^{2} + 2 a^{3} b^{3} c d - b^{6} c^{2}, \left ( t \mapsto t \log{\left (x + \frac{18 t^{2} b c^{2} d^{2} + 3 t a^{3} c d^{2} - 12 t b^{3} c^{2} d + 7 a^{3} b^{2} c d + 2 b^{5} c^{2}}{a^{5} d^{2} + 8 a^{2} b^{3} c d} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*c**2*d**3 - 27*_t**2*b**2*c**2*d**2 + _t*(18*a**3*b*c*d**2 + 9*
b**4*c**2*d) - a**6*d**2 + 2*a**3*b**3*c*d - b**6*c**2, Lambda(_t, _t*log(x + (1
8*_t**2*b*c**2*d**2 + 3*_t*a**3*c*d**2 - 12*_t*b**3*c**2*d + 7*a**3*b**2*c*d + 2
*b**5*c**2)/(a**5*d**2 + 8*a**2*b**3*c*d))))

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GIAC/XCAS [A]  time = 0.21605, size = 262, normalized size = 1.41 \[ \frac{b^{2}{\rm ln}\left ({\left | d x^{3} + c \right |}\right )}{3 \, d} - \frac{{\left (2 \, a b d \left (-\frac{c}{d}\right )^{\frac{1}{3}} + a^{2} d\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d} + \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d - 2 \, \left (-c d^{2}\right )^{\frac{2}{3}} a b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \, c d^{2}} + \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} a^{2} c d^{3} + 2 \, \left (-c d^{2}\right )^{\frac{2}{3}} a b c d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \, c^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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