Integrand size = 17, antiderivative size = 186 \[ \int \frac {(a+b x)^2}{c+d x^3} \, dx=-\frac {a \left (2 b \sqrt [3]{c}+a \sqrt [3]{d}\right ) \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} d^{2/3}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d} \] Output:
-1/3*a*(2*b*c^(1/3)+a*d^(1/3))*arctan(1/3*(c^(1/3)-2*d^(1/3)*x)*3^(1/2)/c^ (1/3))*3^(1/2)/c^(2/3)/d^(2/3)-1/3*a*(2*b*c^(1/3)-a*d^(1/3))*ln(c^(1/3)+d^ (1/3)*x)/c^(2/3)/d^(2/3)+1/6*a*(2*b*c^(1/3)-a*d^(1/3))*ln(c^(2/3)-c^(1/3)* d^(1/3)*x+d^(2/3)*x^2)/c^(2/3)/d^(2/3)+1/3*b^2*ln(d*x^3+c)/d
Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^2}{c+d x^3} \, dx=\frac {\left (2 a b c^{2/3}+a^2 \sqrt [3]{c} \sqrt [3]{d}\right ) \arctan \left (\frac {-\sqrt [3]{c}+2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c d^{2/3}}+\frac {\left (-2 a b c^{2/3}+a^2 \sqrt [3]{c} \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c d^{2/3}}-\frac {\left (-2 a b c^{2/3}+a^2 \sqrt [3]{c} \sqrt [3]{d}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d} \] Input:
Integrate[(a + b*x)^2/(c + d*x^3),x]
Output:
((2*a*b*c^(2/3) + a^2*c^(1/3)*d^(1/3))*ArcTan[(-c^(1/3) + 2*d^(1/3)*x)/(Sq rt[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)
Time = 0.75 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {2410, 792, 2399, 16, 27, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b x)^2}{c+d x^3} \, dx\) |
| \(\Big \downarrow \) 2410 |
| \(\displaystyle \int \frac {a^2+2 b x a}{d x^3+c}dx+b^2 \int \frac {x^2}{d x^3+c}dx\) |
| \(\Big \downarrow \) 792 |
| \(\displaystyle \int \frac {a^2+2 b x a}{d x^3+c}dx+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\int \frac {a \left (2 \sqrt [3]{c} \left (\sqrt [3]{d} a+b \sqrt [3]{c}\right )+\left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3} \sqrt [3]{d}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\int \frac {a \left (2 \sqrt [3]{c} \left (\sqrt [3]{d} a+b \sqrt [3]{c}\right )+\left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {2 \sqrt [3]{c} \left (\sqrt [3]{d} a+b \sqrt [3]{c}\right )+\left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {a \left (\frac {3}{2} \sqrt [3]{c} \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {1}{2} \left (a-\frac {2 b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (\frac {3}{2} \sqrt [3]{c} \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \left (a-\frac {2 b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {3}{2} \sqrt [3]{c} \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \sqrt [3]{d} \left (a-\frac {2 b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {a \left (\frac {1}{2} \sqrt [3]{d} \left (a-\frac {2 b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {a \left (\frac {1}{2} \sqrt [3]{d} \left (a-\frac {2 b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {a \left (-\frac {\sqrt {3} \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {1}{2} \left (a-\frac {2 b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}+\frac {b^2 \log \left (c+d x^3\right )}{3 d}\) |
Input:
Int[(a + b*x)^2/(c + d*x^3),x]
Output:
-1/3*(a*(2*b*c^(1/3) - a*d^(1/3))*Log[c^(1/3) + d^(1/3)*x])/(c^(2/3)*d^(2/ 3)) + (a*(-((Sqrt[3]*(2*b*c^(1/3) + a*d^(1/3))*ArcTan[(1 - (2*d^(1/3)*x)/c ^(1/3))/Sqrt[3]])/d^(1/3)) - ((a - (2*b*c^(1/3))/d^(1/3))*Log[c^(2/3) - c^ (1/3)*d^(1/3)*x + d^(2/3)*x^2])/2))/(3*c^(2/3)*d^(1/3)) + (b^2*Log[c + d*x ^3])/(3*d)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numer ator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Simp[(-r)*((B*r - A*s)/(3*a *s)) Int[1/(r + s*x), x], x] + Simp[r/(3*a*s) Int[(r*(B*r + 2*A*s) + s* (B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] & & NeQ[a*B^3 - b*A^3, 0] && PosQ[a/b]
Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, Int[(A + B*x)/(a + b*x^3), x] + Si mp[C Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !RationalQ[ a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.23
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} d +c \right )}{\sum }\frac {\left (\textit {\_R}^{2} b^{2}+2 \textit {\_R} a b +a^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 d}\) | \(43\) |
| default | \(a^{2} \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )+2 a b \left (-\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )+\frac {b^{2} \ln \left (d \,x^{3}+c \right )}{3 d}\) | \(206\) |
Input:
int((b*x+a)^2/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
1/3/d*sum((_R^2*b^2+2*_R*a*b+a^2)/_R^2*ln(x-_R),_R=RootOf(_Z^3*d+c))
Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 5014, normalized size of antiderivative = 26.96 \[ \int \frac {(a+b x)^2}{c+d x^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^2/(d*x^3+c),x, algorithm="fricas")
Output:
Too large to include
Time = 0.59 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^2}{c+d x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} c^{2} d^{3} - 27 t^{2} b^{2} c^{2} d^{2} + t \left (18 a^{3} b c d^{2} + 9 b^{4} c^{2} d\right ) - a^{6} d^{2} + 2 a^{3} b^{3} c d - b^{6} c^{2}, \left ( t \mapsto t \log {\left (x + \frac {18 t^{2} b c^{2} d^{2} + 3 t a^{3} c d^{2} - 12 t b^{3} c^{2} d + 7 a^{3} b^{2} c d + 2 b^{5} c^{2}}{a^{5} d^{2} + 8 a^{2} b^{3} c d} \right )} \right )\right )} \] Input:
integrate((b*x+a)**2/(d*x**3+c),x)
Output:
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 + (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))))
Time = 0.11 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^2}{c+d x^3} \, dx=-\frac {\sqrt {3} {\left (2 \, b^{2} c - {\left (6 \, a b \left (\frac {c}{d}\right )^{\frac {2}{3}} + 3 \, a^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} + \frac {2 \, b^{2} c}{d}\right )} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{9 \, c d} + \frac {{\left (2 \, b^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} + 2 \, a b \left (\frac {c}{d}\right )^{\frac {1}{3}} - a^{2}\right )} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, d \left (\frac {c}{d}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 2 \, a b \left (\frac {c}{d}\right )^{\frac {1}{3}} + a^{2}\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, d \left (\frac {c}{d}\right )^{\frac {2}{3}}} \] Input:
integrate((b*x+a)^2/(d*x^3+c),x, algorithm="maxima")
Output:
-1/9*sqrt(3)*(2*b^2*c - (6*a*b*(c/d)^(2/3) + 3*a^2*(c/d)^(1/3) + 2*b^2*c/d )*d)*arctan(1/3*sqrt(3)*(2*x - (c/d)^(1/3))/(c/d)^(1/3))/(c*d) + 1/6*(2*b^ 2*(c/d)^(2/3) + 2*a*b*(c/d)^(1/3) - a^2)*log(x^2 - x*(c/d)^(1/3) + (c/d)^( 2/3))/(d*(c/d)^(2/3)) + 1/3*(b^2*(c/d)^(2/3) - 2*a*b*(c/d)^(1/3) + a^2)*lo g(x + (c/d)^(1/3))/(d*(c/d)^(2/3))
Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^2}{c+d x^3} \, dx=\frac {b^{2} \log \left ({\left | d x^{3} + c \right |}\right )}{3 \, d} - \frac {\sqrt {3} {\left (a^{2} d - 2 \, \left (-c d^{2}\right )^{\frac {1}{3}} a b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-c d^{2}\right )^{\frac {2}{3}}} - \frac {{\left (a^{2} d + 2 \, \left (-c d^{2}\right )^{\frac {1}{3}} a b\right )} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, \left (-c d^{2}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, a b d \left (-\frac {c}{d}\right )^{\frac {1}{3}} + a^{2} d\right )} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, c d} \] Input:
integrate((b*x+a)^2/(d*x^3+c),x, algorithm="giac")
Output:
1/3*b^2*log(abs(d*x^3 + c))/d - 1/3*sqrt(3)*(a^2*d - 2*(-c*d^2)^(1/3)*a*b) *arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(-c*d^2)^(2/3) - 1/ 6*(a^2*d + 2*(-c*d^2)^(1/3)*a*b)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/ (-c*d^2)^(2/3) - 1/3*(2*a*b*d*(-c/d)^(1/3) + a^2*d)*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(c*d)
Time = 5.99 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b x)^2}{c+d x^3} \, dx=\sum _{k=1}^3\ln \left (b^4\,c+{\mathrm {root}\left (27\,c^2\,d^3\,z^3-27\,b^2\,c^2\,d^2\,z^2+18\,a^3\,b\,c\,d^2\,z+9\,b^4\,c^2\,d\,z+2\,a^3\,b^3\,c\,d-b^6\,c^2-a^6\,d^2,z,k\right )}^2\,c\,d^2\,9+2\,a^3\,b\,d-\mathrm {root}\left (27\,c^2\,d^3\,z^3-27\,b^2\,c^2\,d^2\,z^2+18\,a^3\,b\,c\,d^2\,z+9\,b^4\,c^2\,d\,z+2\,a^3\,b^3\,c\,d-b^6\,c^2-a^6\,d^2,z,k\right )\,b^2\,c\,d\,6+\mathrm {root}\left (27\,c^2\,d^3\,z^3-27\,b^2\,c^2\,d^2\,z^2+18\,a^3\,b\,c\,d^2\,z+9\,b^4\,c^2\,d\,z+2\,a^3\,b^3\,c\,d-b^6\,c^2-a^6\,d^2,z,k\right )\,a^2\,d^2\,x\,3+3\,a^2\,b^2\,d\,x\right )\,\mathrm {root}\left (27\,c^2\,d^3\,z^3-27\,b^2\,c^2\,d^2\,z^2+18\,a^3\,b\,c\,d^2\,z+9\,b^4\,c^2\,d\,z+2\,a^3\,b^3\,c\,d-b^6\,c^2-a^6\,d^2,z,k\right ) \] Input:
int((a + b*x)^2/(c + d*x^3),x)
Output:
symsum(log(b^4*c + 9*root(27*c^2*d^3*z^3 - 27*b^2*c^2*d^2*z^2 + 18*a^3*b*c *d^2*z + 9*b^4*c^2*d*z + 2*a^3*b^3*c*d - b^6*c^2 - a^6*d^2, z, k)^2*c*d^2 + 2*a^3*b*d - 6*root(27*c^2*d^3*z^3 - 27*b^2*c^2*d^2*z^2 + 18*a^3*b*c*d^2* z + 9*b^4*c^2*d*z + 2*a^3*b^3*c*d - b^6*c^2 - a^6*d^2, z, k)*b^2*c*d + 3*r oot(27*c^2*d^3*z^3 - 27*b^2*c^2*d^2*z^2 + 18*a^3*b*c*d^2*z + 9*b^4*c^2*d*z + 2*a^3*b^3*c*d - b^6*c^2 - a^6*d^2, z, k)*a^2*d^2*x + 3*a^2*b^2*d*x)*roo t(27*c^2*d^3*z^3 - 27*b^2*c^2*d^2*z^2 + 18*a^3*b*c*d^2*z + 9*b^4*c^2*d*z + 2*a^3*b^3*c*d - b^6*c^2 - a^6*d^2, z, k), k, 1, 3)
Time = 0.15 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^2}{c+d x^3} \, dx=\frac {-2 d^{\frac {4}{3}} c^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}-4 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) a b c d -d^{\frac {4}{3}} c^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) a^{2}+2 d^{\frac {4}{3}} c^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) a^{2}+2 d^{\frac {2}{3}} c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) b^{2}+2 d^{\frac {2}{3}} c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) b^{2}+2 \,\mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) a b c d -4 \,\mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) a b c d}{6 d^{\frac {5}{3}} c^{\frac {4}{3}}} \] Input:
int((b*x+a)^2/(d*x^3+c),x)
Output:
( - 2*d**(1/3)*c**(2/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*s qrt(3)))*a**2*d - 4*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt( 3)))*a*b*c*d - d**(1/3)*c**(2/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**( 2/3)*x**2)*a**2*d + 2*d**(1/3)*c**(2/3)*log(c**(1/3) + d**(1/3)*x)*a**2*d + 2*d**(2/3)*c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)* b**2*c + 2*d**(2/3)*c**(1/3)*log(c**(1/3) + d**(1/3)*x)*b**2*c + 2*log(c** (2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*a*b*c*d - 4*log(c**(1/3) + d* *(1/3)*x)*a*b*c*d)/(6*d**(2/3)*c**(1/3)*c*d)