Integrand size = 32, antiderivative size = 371 \[ \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\frac {3 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt [3]{c+d x}}{b^4}+\frac {3 \left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) (c+d x)^{4/3}}{4 b^3 d^3}+\frac {3 (b C d-2 b c D-a d D) (c+d x)^{7/3}}{7 b^2 d^3}+\frac {3 D (c+d x)^{10/3}}{10 b d^3}-\frac {\sqrt {3} \sqrt [3]{b c-a d} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{b^{13/3}}-\frac {\sqrt [3]{b c-a d} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log (a+b x)}{2 b^{13/3}}+\frac {3 \sqrt [3]{b c-a d} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{13/3}} \] Output:
3*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(1/3)/b^4+3/4*(a^2*d^2*D-a*b*d*(C* d-D*c)-b^2*(-B*d^2+C*c*d-D*c^2))*(d*x+c)^(4/3)/b^3/d^3+3/7*(C*b*d-D*a*d-2* D*b*c)*(d*x+c)^(7/3)/b^2/d^3+3/10*D*(d*x+c)^(10/3)/b/d^3-3^(1/2)*(-a*d+b*c )^(1/3)*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*arctan(1/3*(1+2*b^(1/3)*(d*x+c)^(1/3 )/(-a*d+b*c)^(1/3))*3^(1/2))/b^(13/3)-1/2*(-a*d+b*c)^(1/3)*(A*b^3-a*(B*b^2 -C*a*b+D*a^2))*ln(b*x+a)/b^(13/3)+3/2*(-a*d+b*c)^(1/3)*(A*b^3-a*(B*b^2-C*a *b+D*a^2))*ln((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))/b^(13/3)
Time = 0.63 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\frac {\frac {3 \sqrt [3]{b} \sqrt [3]{c+d x} \left (-140 a^3 d^3 D+35 a^2 b d^2 (4 C d+D (c+d x))-5 a b^2 d \left (-3 c^2 D+c d (7 C+D x)+d^2 \left (28 B+7 C x+4 D x^2\right )\right )+b^3 \left (9 c^3 D-3 c^2 d (5 C+D x)+c d^2 \left (35 B+5 C x+2 D x^2\right )+d^3 \left (140 A+35 B x+20 C x^2+14 D x^3\right )\right )\right )}{d^3}+140 \sqrt {3} \sqrt [3]{-b c+a d} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{-b c+a d}}}{\sqrt {3}}\right )-140 \sqrt [3]{-b c+a d} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log \left (\sqrt [3]{-b c+a d}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )+70 \sqrt [3]{-b c+a d} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log \left ((-b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b c+a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{140 b^{13/3}} \] Input:
Integrate[((c + d*x)^(1/3)*(A + B*x + C*x^2 + D*x^3))/(a + b*x),x]
Output:
((3*b^(1/3)*(c + d*x)^(1/3)*(-140*a^3*d^3*D + 35*a^2*b*d^2*(4*C*d + D*(c + d*x)) - 5*a*b^2*d*(-3*c^2*D + c*d*(7*C + D*x) + d^2*(28*B + 7*C*x + 4*D*x ^2)) + b^3*(9*c^3*D - 3*c^2*d*(5*C + D*x) + c*d^2*(35*B + 5*C*x + 2*D*x^2) + d^3*(140*A + 35*B*x + 20*C*x^2 + 14*D*x^3))))/d^3 + 140*Sqrt[3]*(-(b*c) + a*d)^(1/3)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*ArcTan[(1 - (2*b^(1/3)*( c + d*x)^(1/3))/(-(b*c) + a*d)^(1/3))/Sqrt[3]] - 140*(-(b*c) + a*d)^(1/3)* (A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Log[(-(b*c) + a*d)^(1/3) + b^(1/3)*(c + d*x)^(1/3)] + 70*(-(b*c) + a*d)^(1/3)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D) )*Log[(-(b*c) + a*d)^(2/3) - b^(1/3)*(-(b*c) + a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3)])/(140*b^(13/3))
Time = 0.71 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2123, 2009}
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.
| \(\displaystyle \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {\sqrt [3]{c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^3 (a+b x)}+\frac {\sqrt [3]{c+d x} \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^2}+\frac {(c+d x)^{4/3} (-a d D-2 b c D+b C d)}{b^2 d^2}+\frac {D (c+d x)^{7/3}}{b d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {3} \sqrt [3]{b c-a d} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt {3}}\right )}{b^{13/3}}-\frac {\sqrt [3]{b c-a d} \log (a+b x) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{2 b^{13/3}}+\frac {3 \sqrt [3]{b c-a d} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{13/3}}+\frac {3 \sqrt [3]{c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^4}+\frac {3 (c+d x)^{4/3} \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{4 b^3 d^3}+\frac {3 (c+d x)^{7/3} (-a d D-2 b c D+b C d)}{7 b^2 d^3}+\frac {3 D (c+d x)^{10/3}}{10 b d^3}\) |
Input:
Int[((c + d*x)^(1/3)*(A + B*x + C*x^2 + D*x^3))/(a + b*x),x]
Output:
(3*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(1/3))/b^4 + (3*(a^2*d^2* D - a*b*d*(C*d - c*D) - b^2*(c*C*d - B*d^2 - c^2*D))*(c + d*x)^(4/3))/(4*b ^3*d^3) + (3*(b*C*d - 2*b*c*D - a*d*D)*(c + d*x)^(7/3))/(7*b^2*d^3) + (3*D *(c + d*x)^(10/3))/(10*b*d^3) - (Sqrt[3]*(b*c - a*d)^(1/3)*(A*b^3 - a*(b^2 *B - a*b*C + a^2*D))*ArcTan[(1 + (2*b^(1/3)*(c + d*x)^(1/3))/(b*c - a*d)^( 1/3))/Sqrt[3]])/b^(13/3) - ((b*c - a*d)^(1/3)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Log[a + b*x])/(2*b^(13/3)) + (3*(b*c - a*d)^(1/3)*(A*b^3 - a*(b^2* B - a*b*C + a^2*D))*Log[(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)])/(2*b ^(13/3))
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 1.24 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.92
| method | result | size |
| pseudoelliptic | \(-\frac {-3 \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}} \left (\left (\left (\frac {1}{10} D x^{3}+\frac {1}{7} C \,x^{2}+\frac {1}{4} B x +A \right ) b^{3}-\left (\frac {1}{7} D x^{2}+\frac {1}{4} C x +B \right ) a \,b^{2}+a^{2} \left (\frac {D x}{4}+C \right ) b -a^{3} D\right ) d^{3}+\frac {\left (\left (\frac {2}{35} D x^{2}+\frac {1}{7} C x +B \right ) b^{2}-\left (\frac {D x}{7}+C \right ) a b +D a^{2}\right ) c b \,d^{2}}{4}-\frac {3 \left (\left (\frac {D x}{5}+C \right ) b -D a \right ) c^{2} b^{2} d}{28}+\frac {9 D b^{3} c^{3}}{140}\right ) \left (x d +c \right )^{\frac {1}{3}} b +\left (\arctan \left (\frac {\sqrt {3}\, \left (2 \left (x d +c \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2}\right ) \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \left (a d -b c \right ) d^{3}}{\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}} d^{3} b^{5}}\) | \(340\) |
| derivativedivides | \(\frac {\frac {3 \left (\frac {D \left (x d +c \right )^{\frac {10}{3}} b^{3}}{10}+\frac {C \,b^{3} d \left (x d +c \right )^{\frac {7}{3}}}{7}-\frac {D a \,b^{2} d \left (x d +c \right )^{\frac {7}{3}}}{7}-\frac {2 D b^{3} c \left (x d +c \right )^{\frac {7}{3}}}{7}+\frac {B \,b^{3} d^{2} \left (x d +c \right )^{\frac {4}{3}}}{4}-\frac {C a \,b^{2} d^{2} \left (x d +c \right )^{\frac {4}{3}}}{4}-\frac {C \,b^{3} c d \left (x d +c \right )^{\frac {4}{3}}}{4}+\frac {D a^{2} b \,d^{2} \left (x d +c \right )^{\frac {4}{3}}}{4}+\frac {D a \,b^{2} c d \left (x d +c \right )^{\frac {4}{3}}}{4}+\frac {D b^{3} c^{2} \left (x d +c \right )^{\frac {4}{3}}}{4}+A \,b^{3} d^{3} \left (x d +c \right )^{\frac {1}{3}}-B a \,b^{2} d^{3} \left (x d +c \right )^{\frac {1}{3}}+C \,a^{2} b \,d^{3} \left (x d +c \right )^{\frac {1}{3}}-D a^{3} d^{3} \left (x d +c \right )^{\frac {1}{3}}\right )}{b^{4}}-\frac {3 \left (\frac {\ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\right ) d^{3} \left (A a \,b^{3} d -A \,b^{4} c -B \,a^{2} b^{2} d +B a \,b^{3} c +C \,a^{3} b d -C \,a^{2} b^{2} c -D a^{4} d +D a^{3} b c \right )}{b^{4}}}{d^{3}}\) | \(461\) |
| default | \(\frac {\frac {3 \left (\frac {D \left (x d +c \right )^{\frac {10}{3}} b^{3}}{10}+\frac {C \,b^{3} d \left (x d +c \right )^{\frac {7}{3}}}{7}-\frac {D a \,b^{2} d \left (x d +c \right )^{\frac {7}{3}}}{7}-\frac {2 D b^{3} c \left (x d +c \right )^{\frac {7}{3}}}{7}+\frac {B \,b^{3} d^{2} \left (x d +c \right )^{\frac {4}{3}}}{4}-\frac {C a \,b^{2} d^{2} \left (x d +c \right )^{\frac {4}{3}}}{4}-\frac {C \,b^{3} c d \left (x d +c \right )^{\frac {4}{3}}}{4}+\frac {D a^{2} b \,d^{2} \left (x d +c \right )^{\frac {4}{3}}}{4}+\frac {D a \,b^{2} c d \left (x d +c \right )^{\frac {4}{3}}}{4}+\frac {D b^{3} c^{2} \left (x d +c \right )^{\frac {4}{3}}}{4}+A \,b^{3} d^{3} \left (x d +c \right )^{\frac {1}{3}}-B a \,b^{2} d^{3} \left (x d +c \right )^{\frac {1}{3}}+C \,a^{2} b \,d^{3} \left (x d +c \right )^{\frac {1}{3}}-D a^{3} d^{3} \left (x d +c \right )^{\frac {1}{3}}\right )}{b^{4}}-\frac {3 \left (\frac {\ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\right ) d^{3} \left (A a \,b^{3} d -A \,b^{4} c -B \,a^{2} b^{2} d +B a \,b^{3} c +C \,a^{3} b d -C \,a^{2} b^{2} c -D a^{4} d +D a^{3} b c \right )}{b^{4}}}{d^{3}}\) | \(461\) |
Input:
int((d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/((a*d-b*c)/b)^(2/3)*(-3*((a*d-b*c)/b)^(2/3)*(((1/10*D*x^3+1/7*C*x^2+1/4 *B*x+A)*b^3-(1/7*D*x^2+1/4*C*x+B)*a*b^2+a^2*(1/4*D*x+C)*b-a^3*D)*d^3+1/4*( (2/35*D*x^2+1/7*C*x+B)*b^2-(1/7*D*x+C)*a*b+D*a^2)*c*b*d^2-3/28*((1/5*D*x+C )*b-D*a)*c^2*b^2*d+9/140*D*b^3*c^3)*(d*x+c)^(1/3)*b+(arctan(1/3*3^(1/2)*(2 *(d*x+c)^(1/3)-((a*d-b*c)/b)^(1/3))/((a*d-b*c)/b)^(1/3))*3^(1/2)+ln((d*x+c )^(1/3)+((a*d-b*c)/b)^(1/3))-1/2*ln((d*x+c)^(2/3)-((a*d-b*c)/b)^(1/3)*(d*x +c)^(1/3)+((a*d-b*c)/b)^(2/3)))*(A*b^3-B*a*b^2+C*a^2*b-D*a^3)*(a*d-b*c)*d^ 3)/d^3/b^5
Time = 0.09 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=-\frac {140 \, \sqrt {3} {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} d^{3} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (d x + c\right )}^{\frac {1}{3}} b \left (\frac {b c - a d}{b}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) - 70 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} d^{3} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} + {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{3}}\right ) + 140 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} d^{3} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \log \left ({\left (d x + c\right )}^{\frac {1}{3}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}\right ) - 3 \, {\left (14 \, D b^{3} d^{3} x^{3} + 9 \, D b^{3} c^{3} + 15 \, {\left (D a b^{2} - C b^{3}\right )} c^{2} d + 35 \, {\left (D a^{2} b - C a b^{2} + B b^{3}\right )} c d^{2} - 140 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} d^{3} + 2 \, {\left (D b^{3} c d^{2} - 10 \, {\left (D a b^{2} - C b^{3}\right )} d^{3}\right )} x^{2} - {\left (3 \, D b^{3} c^{2} d + 5 \, {\left (D a b^{2} - C b^{3}\right )} c d^{2} - 35 \, {\left (D a^{2} b - C a b^{2} + B b^{3}\right )} d^{3}\right )} x\right )} {\left (d x + c\right )}^{\frac {1}{3}}}{140 \, b^{4} d^{3}} \] Input:
integrate((d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x, algorithm="fricas")
Output:
-1/140*(140*sqrt(3)*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*d^3*((b*c - a*d)/b )^(1/3)*arctan(-1/3*(2*sqrt(3)*(d*x + c)^(1/3)*b*((b*c - a*d)/b)^(2/3) + s qrt(3)*(b*c - a*d))/(b*c - a*d)) - 70*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)* d^3*((b*c - a*d)/b)^(1/3)*log((d*x + c)^(2/3) + (d*x + c)^(1/3)*((b*c - a* d)/b)^(1/3) + ((b*c - a*d)/b)^(2/3)) + 140*(D*a^3 - C*a^2*b + B*a*b^2 - A* b^3)*d^3*((b*c - a*d)/b)^(1/3)*log((d*x + c)^(1/3) - ((b*c - a*d)/b)^(1/3) ) - 3*(14*D*b^3*d^3*x^3 + 9*D*b^3*c^3 + 15*(D*a*b^2 - C*b^3)*c^2*d + 35*(D *a^2*b - C*a*b^2 + B*b^3)*c*d^2 - 140*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)* d^3 + 2*(D*b^3*c*d^2 - 10*(D*a*b^2 - C*b^3)*d^3)*x^2 - (3*D*b^3*c^2*d + 5* (D*a*b^2 - C*b^3)*c*d^2 - 35*(D*a^2*b - C*a*b^2 + B*b^3)*d^3)*x)*(d*x + c) ^(1/3))/(b^4*d^3)
\[ \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\int \frac {\sqrt [3]{c + d x} \left (A + B x + C x^{2} + D x^{3}\right )}{a + b x}\, dx \] Input:
integrate((d*x+c)**(1/3)*(D*x**3+C*x**2+B*x+A)/(b*x+a),x)
Output:
Integral((c + d*x)**(1/3)*(A + B*x + C*x**2 + D*x**3)/(a + b*x), x)
Exception generated. \[ \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (323) = 646\).
Time = 0.18 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x, algorithm="giac")
Output:
-(D*a^3*b^7*c*d^33 - C*a^2*b^8*c*d^33 + B*a*b^9*c*d^33 - A*b^10*c*d^33 - D *a^4*b^6*d^34 + C*a^3*b^7*d^34 - B*a^2*b^8*d^34 + A*a*b^9*d^34)*((b*c - a* d)/b)^(1/3)*log(abs((d*x + c)^(1/3) - ((b*c - a*d)/b)^(1/3)))/(b^11*c*d^33 - a*b^10*d^34) + (sqrt(3)*(b^3*c - a*b^2*d)^(1/3)*D*a^3 - sqrt(3)*(b^3*c - a*b^2*d)^(1/3)*C*a^2*b + sqrt(3)*(b^3*c - a*b^2*d)^(1/3)*B*a*b^2 - sqrt( 3)*(b^3*c - a*b^2*d)^(1/3)*A*b^3)*arctan(1/3*sqrt(3)*(2*(d*x + c)^(1/3) + ((b*c - a*d)/b)^(1/3))/((b*c - a*d)/b)^(1/3))/b^5 + 1/2*((b^3*c - a*b^2*d) ^(1/3)*D*a^3 - (b^3*c - a*b^2*d)^(1/3)*C*a^2*b + (b^3*c - a*b^2*d)^(1/3)*B *a*b^2 - (b^3*c - a*b^2*d)^(1/3)*A*b^3)*log((d*x + c)^(2/3) + (d*x + c)^(1 /3)*((b*c - a*d)/b)^(1/3) + ((b*c - a*d)/b)^(2/3))/b^5 + 3/140*(14*(d*x + c)^(10/3)*D*b^9*d^27 - 40*(d*x + c)^(7/3)*D*b^9*c*d^27 + 35*(d*x + c)^(4/3 )*D*b^9*c^2*d^27 - 20*(d*x + c)^(7/3)*D*a*b^8*d^28 + 20*(d*x + c)^(7/3)*C* b^9*d^28 + 35*(d*x + c)^(4/3)*D*a*b^8*c*d^28 - 35*(d*x + c)^(4/3)*C*b^9*c* d^28 + 35*(d*x + c)^(4/3)*D*a^2*b^7*d^29 - 35*(d*x + c)^(4/3)*C*a*b^8*d^29 + 35*(d*x + c)^(4/3)*B*b^9*d^29 - 140*(d*x + c)^(1/3)*D*a^3*b^6*d^30 + 14 0*(d*x + c)^(1/3)*C*a^2*b^7*d^30 - 140*(d*x + c)^(1/3)*B*a*b^8*d^30 + 140* (d*x + c)^(1/3)*A*b^9*d^30)/(b^10*d^30)
Timed out. \[ \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\int \frac {{\left (c+d\,x\right )}^{1/3}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{a+b\,x} \,d x \] Input:
int(((c + d*x)^(1/3)*(A + B*x + C*x^2 + x^3*D))/(a + b*x),x)
Output:
int(((c + d*x)^(1/3)*(A + B*x + C*x^2 + x^3*D))/(a + b*x), x)
Time = 0.17 (sec) , antiderivative size = 1132, normalized size of antiderivative = 3.05 \[ \int \frac {\sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x)
Output:
( - 140*sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) - 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b*c)**(1/6)))*a**4*d**4 + 280*sqrt(3)*atan ((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) - 2*b**(1/3)*(c + d*x)**(1/6))/(b**( 1/6)*(a*d - b*c)**(1/6)))*a**3*b*c*d**3 - 140*sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) - 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b*c) **(1/6)))*a**2*b**2*c**2*d**2 - 140*sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1 /6)*sqrt(3) + 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b*c)**(1/6)))* a**4*d**4 + 280*sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) + 2*b**( 1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b*c)**(1/6)))*a**3*b*c*d**3 - 140* sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) + 2*b**(1/3)*(c + d*x)** (1/6))/(b**(1/6)*(a*d - b*c)**(1/6)))*a**2*b**2*c**2*d**2 - 420*b**(1/3)*( c + d*x)**(1/3)*(a*d - b*c)**(2/3)*a**3*d**3 + 525*b**(1/3)*(c + d*x)**(1/ 3)*(a*d - b*c)**(2/3)*a**2*b*c*d**2 + 105*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*a**2*b*d**3*x - 60*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/ 3)*a*b**2*c**2*d - 120*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*a*b**2 *c*d**2*x - 60*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*a*b**2*d**3*x* *2 + 105*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*b**4*c*d + 105*b**(1 /3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*b**4*d**2*x - 18*b**(1/3)*(c + d*x )**(1/3)*(a*d - b*c)**(2/3)*b**3*c**3 + 6*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*b**3*c**2*d*x + 66*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**...