Integrand size = 60, antiderivative size = 1655 \[ \int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx =\text {Too large to display} \]
(b-x)^(2/3)*(-a+x)^(1/3)*(-1/2*3^(1/2)*a*(-1+d^(1/2))*arctan(3^(1/2)*(-a+x )^(1/3)/(-2*d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)^2/d^(2/3)-1/2*3^(1/2) *b*c*(-1+d^(1/2))*arctan(3^(1/2)*(-a+x)^(1/3)/(-2*d^(1/6)*(b-x)^(1/3)+(-a+ x)^(1/3)))/(a-b)^2/d^(2/3)-1/2*3^(1/2)*(a-b*d^(1/2))*arctan(3^(1/2)*(-a+x) ^(1/3)/(-2*d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)^2/d^(2/3)-1/2*3^(1/2)* c*(a-b*d^(1/2))*arctan(3^(1/2)*(-a+x)^(1/3)/(-2*d^(1/6)*(b-x)^(1/3)+(-a+x) ^(1/3)))/(a-b)^2/d^(2/3)+1/2*3^(1/2)*a*(1+d^(1/2))*arctan(3^(1/2)*(-a+x)^( 1/3)/(2*d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)^2/d^(2/3)+1/2*3^(1/2)*b*c *(1+d^(1/2))*arctan(3^(1/2)*(-a+x)^(1/3)/(2*d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/ 3)))/(a-b)^2/d^(2/3)-1/2*3^(1/2)*(a+b*d^(1/2))*arctan(3^(1/2)*(-a+x)^(1/3) /(2*d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)^2/d^(2/3)-1/2*3^(1/2)*c*(a+b* d^(1/2))*arctan(3^(1/2)*(-a+x)^(1/3)/(2*d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))) /(a-b)^2/d^(2/3)+1/2*a*(1+d^(1/2))*ln(-d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))/( a-b)^2/d^(2/3)+1/2*b*c*(1+d^(1/2))*ln(-d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))/( a-b)^2/d^(2/3)+1/2*(-a-b*d^(1/2))*ln(-d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))/(a -b)^2/d^(2/3)-1/2*c*(a+b*d^(1/2))*ln(-d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))/(a -b)^2/d^(2/3)-1/2*a*(-1+d^(1/2))*ln(d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b )^2/d^(2/3)-1/2*b*c*(-1+d^(1/2))*ln(d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b )^2/d^(2/3)-1/2*c*(a-b*d^(1/2))*ln(d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b) ^2/d^(2/3)+1/2*(-a+b*d^(1/2))*ln(d^(1/6)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b...
Time = 0.44 (sec) , antiderivative size = 501, normalized size of antiderivative = 0.30 \[ \int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \left (c+\sqrt {d}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (c-\sqrt {d}\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )+c \log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\sqrt {d} \log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 c \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+2 \sqrt {d} \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 c \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \sqrt {d} \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+c \log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-\sqrt {d} \log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )\right )}{4 (a-b) d^{2/3} \sqrt [3]{(b-x)^2 (-a+x)}} \]
Integrate[(-a - b*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-a^2 + b^2* d + 2*(a - b*d)*x + (-1 + d)*x^2)),x]
((b - x)^(2/3)*(-a + x)^(1/3)*(2*Sqrt[3]*(c + Sqrt[d])*ArcTan[(1 - (2*d^(1 /6)*(b - x)^(1/3))/(-a + x)^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(c - Sqrt[d])*ArcT an[(1 + (2*d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/3))/Sqrt[3]] + c*Log[1 + (d^ (1/3)*(b - x)^(2/3))/(-a + x)^(2/3) - (d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/ 3)] + Sqrt[d]*Log[1 + (d^(1/3)*(b - x)^(2/3))/(-a + x)^(2/3) - (d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/3)] - 2*c*Log[-1 + (d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/3)] + 2*Sqrt[d]*Log[-1 + (d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/3)] - 2 *c*Log[1 + (d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/3)] - 2*Sqrt[d]*Log[1 + (d^ (1/6)*(b - x)^(1/3))/(-a + x)^(1/3)] + c*Log[1 + (d^(1/3)*(b - x)^(2/3))/( -a + x)^(2/3) + (d^(1/6)*(b - x)^(1/3))/(-a + x)^(1/3)] - Sqrt[d]*Log[1 + (d^(1/3)*(b - x)^(2/3))/(-a + x)^(2/3) + (d^(1/6)*(b - x)^(1/3))/(-a + x)^ (1/3)]))/(4*(a - b)*d^(2/3)*((b - x)^2*(-a + x))^(1/3))
Time = 1.18 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.23, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {7270, 2153, 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.
| \(\displaystyle \int \frac {-a-b c+(c+1) x}{\sqrt [3]{(x-a) (x-b)^2} \left (-a^2+2 x (a-b d)+b^2 d+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [3]{x-a} (x-b)^{2/3} \int \frac {a+b c-(c+1) x}{\sqrt [3]{x-a} (x-b)^{2/3} \left (a^2+(1-d) x^2-b^2 d-2 (a-b d) x\right )}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 2153 |
| \(\displaystyle \frac {\sqrt [3]{x-a} (x-b)^{2/3} \int \left (\frac {-c-\frac {-c-d}{\sqrt {d}}-1}{\sqrt [3]{x-a} (x-b)^{2/3} \left (2 \sqrt {d} (a-b)-2 (a-b d)+2 (1-d) x\right )}+\frac {-c+\frac {-c-d}{\sqrt {d}}-1}{\sqrt [3]{x-a} (x-b)^{2/3} \left (-2 \sqrt {d} (a-b)-2 (a-b d)+2 (1-d) x\right )}\right )dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (-\frac {\sqrt {3} \left (c-\sqrt {d}\right ) \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-b}}\right )}{2 d^{2/3} (a-b)}-\frac {\sqrt {3} \left (c+\sqrt {d}\right ) \arctan \left (\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 d^{2/3} (a-b)}+\frac {\left (c+\sqrt {d}\right ) \log \left (2 (1-d) x-2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )\right )}{4 d^{2/3} (a-b)}+\frac {\left (c-\sqrt {d}\right ) \log \left (2 (1-d) x-2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )\right )}{4 d^{2/3} (a-b)}-\frac {3 \left (c-\sqrt {d}\right ) \log \left (-\frac {\sqrt [3]{x-a}}{\sqrt [6]{d}}-\sqrt [3]{x-b}\right )}{4 d^{2/3} (a-b)}-\frac {3 \left (c+\sqrt {d}\right ) \log \left (\frac {\sqrt [3]{x-a}}{\sqrt [6]{d}}-\sqrt [3]{x-b}\right )}{4 d^{2/3} (a-b)}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
Int[(-a - b*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-a^2 + b^2*d + 2* (a - b*d)*x + (-1 + d)*x^2)),x]
((-a + x)^(1/3)*(-b + x)^(2/3)*(-1/2*(Sqrt[3]*(c - Sqrt[d])*ArcTan[1/Sqrt[ 3] - (2*(-a + x)^(1/3))/(Sqrt[3]*d^(1/6)*(-b + x)^(1/3))])/((a - b)*d^(2/3 )) - (Sqrt[3]*(c + Sqrt[d])*ArcTan[1/Sqrt[3] + (2*(-a + x)^(1/3))/(Sqrt[3] *d^(1/6)*(-b + x)^(1/3))])/(2*(a - b)*d^(2/3)) + ((c + Sqrt[d])*Log[-2*(1 + Sqrt[d])*(a - b*Sqrt[d]) + 2*(1 - d)*x])/(4*(a - b)*d^(2/3)) + ((c - Sqr t[d])*Log[-2*(1 - Sqrt[d])*(a + b*Sqrt[d]) + 2*(1 - d)*x])/(4*(a - b)*d^(2 /3)) - (3*(c - Sqrt[d])*Log[-((-a + x)^(1/3)/d^(1/6)) - (-b + x)^(1/3)])/( 4*(a - b)*d^(2/3)) - (3*(c + Sqrt[d])*Log[(-a + x)^(1/3)/d^(1/6) - (-b + x )^(1/3)])/(4*(a - b)*d^(2/3))))/(-((a - x)*(b - x)^2))^(1/3)
3.32.48.3.1 Defintions of rubi rules used
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ [m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int \frac {-a -b c +\left (1+c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-a^{2}+b^{2} d +2 \left (-b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 5102 vs. \(2 (1277) = 2554\).
Time = 0.46 (sec) , antiderivative size = 5102, normalized size of antiderivative = 3.08 \[ \int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\text {Too large to display} \]
integrate((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-a^2+b^2*d+2*(-b*d+a)* x+(-1+d)*x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)**2)**(1/3)/(-a**2+b**2*d+2*(-b*d +a)*x+(-1+d)*x**2),x)
\[ \int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}} \,d x } \]
integrate((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-a^2+b^2*d+2*(-b*d+a)* x+(-1+d)*x^2),x, algorithm="maxima")
-integrate((b*c - (c + 1)*x + a)/((-(a - x)*(b - x)^2)^(1/3)*(b^2*d + (d - 1)*x^2 - a^2 - 2*(b*d - a)*x)), x)
\[ \int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}} \,d x } \]
integrate((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-a^2+b^2*d+2*(-b*d+a)* x+(-1+d)*x^2),x, algorithm="giac")
integrate(-(b*c - (c + 1)*x + a)/((-(a - x)*(b - x)^2)^(1/3)*(b^2*d + (d - 1)*x^2 - a^2 - 2*(b*d - a)*x)), x)
Timed out. \[ \int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=-\int \frac {a+b\,c-x\,\left (c+1\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (b^2\,d+2\,x\,\left (a-b\,d\right )-a^2+x^2\,\left (d-1\right )\right )} \,d x \]
int(-(a + b*c - x*(c + 1))/((-(a - x)*(b - x)^2)^(1/3)*(b^2*d + 2*x*(a - b *d) - a^2 + x^2*(d - 1))),x)