3.32.35 \(\int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx\) [3135]

Optimal. Leaf size=857 \[ \frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (\frac {3 a \sqrt [3]{b-x} (-a+x)^{2/3}}{2 (a-b)^2 (-b+x)}-\frac {3 b \sqrt [3]{b-x} (-a+x)^{2/3}}{2 (a-b)^2 (-b+x)}-\frac {\sqrt {3} a (-1+d) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} b c (-1+d) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} (a-b d) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} c (a-b d) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {a (-1+d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {b c (-1+d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {c (a-b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}+\frac {(-a+b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}+\frac {a (-1+d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {b c (-1+d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {(a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {c (a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}\right )}{\sqrt [3]{(b-x)^2 (-a+x)}} \]

[Out]

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

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Rubi [A]
time = 1.07, antiderivative size = 283, normalized size of antiderivative = 0.33, number of steps used = 4, number of rules used = 4, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6851, 160, 12, 93} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} (c+d) \text {ArcTan}\left (\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{d} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} (c+d) \log (a-b d-(1-d) x)}{2 \sqrt [3]{d} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} (c+d) \log \left (\frac {\sqrt [3]{x-a}}{\sqrt [3]{d}}-\sqrt [3]{x-b}\right )}{2 \sqrt [3]{d} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 (a-x)}{2 (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-a - b*c + (1 + c)*x)/((-b + x)*((-a + x)*(-b + x)^2)^(1/3)*(a - b*d + (-1 + d)*x)),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])*q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1
/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q*(a + b*x)^(1/3) - (c +
d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {-a-b c+(1+c) x}{\sqrt [3]{-a+x} (-b+x)^{5/3} (a-b d+(-1+d) x)} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {3 (a-x)}{2 (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (3 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {2 (a-b)^2 (c+d)}{3 \sqrt [3]{-a+x} (-b+x)^{2/3} (a-b d+(-1+d) x)} \, dx}{2 (a-b)^2 \sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {3 (a-x)}{2 (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left ((c+d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} (a-b d+(-1+d) x)} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {3 (a-x)}{2 (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} (c+d) \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-b+x}}\right )}{(a-b) \sqrt [3]{d} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {(c+d) \sqrt [3]{-a+x} (-b+x)^{2/3} \log (a-b d-(1-d) x)}{2 (a-b) \sqrt [3]{d} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 (c+d) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (\frac {\sqrt [3]{-a+x}}{\sqrt [3]{d}}-\sqrt [3]{-b+x}\right )}{2 (a-b) \sqrt [3]{d} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 282, normalized size = 0.33 \begin {gather*} \frac {3 a \sqrt [3]{d}-3 \sqrt [3]{d} x+2 \sqrt {3} (c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )+(c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (1+\frac {d^{2/3} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 c (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 d (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )}{2 (a-b) \sqrt [3]{d} \sqrt [3]{(b-x)^2 (-a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-a - b*c + (1 + c)*x)/((-b + x)*((-a + x)*(-b + x)^2)^(1/3)*(a - b*d + (-1 + d)*x)),x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {-a -b c +\left (1+c \right ) x}{\left (-b +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (a -b d +\left (-1+d \right ) x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(a-b*d+(-1+d)*x),x)

[Out]

int((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(a-b*d+(-1+d)*x),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(a-b*d+(-1+d)*x),x, algorithm="maxima")

[Out]

-integrate((b*c - (c + 1)*x + a)/((-(a - x)*(b - x)^2)^(1/3)*(b*d - (d - 1)*x - a)*(b - x)), x)

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Fricas [A]
time = 0.43, size = 976, normalized size = 1.14 \begin {gather*} \left [-\frac {\sqrt {3} {\left (b^{2} c d + b^{2} d^{2} + {\left (c d + d^{2}\right )} x^{2} - 2 \, {\left (b c d + b d^{2}\right )} x\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {b^{2} d + {\left (d + 2\right )} x^{2} + 2 \, a b + 3 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} d^{\frac {2}{3}} - 2 \, {\left (b d + a + b\right )} x + \sqrt {3} {\left ({\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b d - d x\right )} - {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} d^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}}}{b^{2} d + {\left (d - 1\right )} x^{2} - a b - {\left (2 \, b d - a - b\right )} x}\right ) - {\left (b^{2} c + b^{2} d + {\left (c + d\right )} x^{2} - 2 \, {\left (b c + b d\right )} x\right )} d^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} d^{\frac {1}{3}} - {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) + 2 \, {\left (b^{2} c + b^{2} d + {\left (c + d\right )} x^{2} - 2 \, {\left (b c + b d\right )} x\right )} d^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} d^{\frac {1}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}}}{b - x}\right ) + 3 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d}{2 \, {\left ({\left (a - b\right )} d x^{2} - 2 \, {\left (a b - b^{2}\right )} d x + {\left (a b^{2} - b^{3}\right )} d\right )}}, \frac {{\left (b^{2} c + b^{2} d + {\left (c + d\right )} x^{2} - 2 \, {\left (b c + b d\right )} x\right )} d^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} d^{\frac {1}{3}} - {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, {\left (b^{2} c + b^{2} d + {\left (c + d\right )} x^{2} - 2 \, {\left (b c + b d\right )} x\right )} d^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} d^{\frac {1}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}}}{b - x}\right ) - \frac {2 \, \sqrt {3} {\left (b^{2} c d + b^{2} d^{2} + {\left (c d + d^{2}\right )} x^{2} - 2 \, {\left (b c d + b d^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {3} {\left ({\left (b - x\right )} d^{\frac {1}{3}} - 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (b - x\right )} d^{\frac {1}{3}}}\right )}{d^{\frac {1}{3}}} - 3 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d}{2 \, {\left ({\left (a - b\right )} d x^{2} - 2 \, {\left (a b - b^{2}\right )} d x + {\left (a b^{2} - b^{3}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(a-b*d+(-1+d)*x),x, algorithm="fricas")

[Out]

[-1/2*(sqrt(3)*(b^2*c*d + b^2*d^2 + (c*d + d^2)*x^2 - 2*(b*c*d + b*d^2)*x)*sqrt(-1/d^(2/3))*log(-(b^2*d + (d +
2)*x^2 + 2*a*b + 3*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b - x)*d^(2/3) - 2*(b*d + a + b)*x
+ sqrt(3)*((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b*d - d*x) - (b^2*d - 2*b*d*x + d*x^2)*d^(
1/3) + 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d^(2/3))*sqrt(-1/d^(2/3)))/(b^2*d + (d - 1)*x^
2 - a*b - (2*b*d - a - b)*x)) - (b^2*c + b^2*d + (c + d)*x^2 - 2*(b*c + b*d)*x)*d^(2/3)*log(-((-a*b^2 - (a + 2
*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b - x)*d^(1/3) - (b^2 - 2*b*x + x^2)*d^(2/3) - (-a*b^2 - (a + 2*b)*x^2
+ x^3 + (2*a*b + b^2)*x)^(2/3))/(b^2 - 2*b*x + x^2)) + 2*(b^2*c + b^2*d + (c + d)*x^2 - 2*(b*c + b*d)*x)*d^(2
/3)*log(-((b - x)*d^(1/3) + (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x)) + 3*(-a*b^2 - (a
+ 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d)/((a - b)*d*x^2 - 2*(a*b - b^2)*d*x + (a*b^2 - b^3)*d), 1/2*((b^2*
c + b^2*d + (c + d)*x^2 - 2*(b*c + b*d)*x)*d^(2/3)*log(-((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3
)*(b - x)*d^(1/3) - (b^2 - 2*b*x + x^2)*d^(2/3) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3))/(b^2
- 2*b*x + x^2)) - 2*(b^2*c + b^2*d + (c + d)*x^2 - 2*(b*c + b*d)*x)*d^(2/3)*log(-((b - x)*d^(1/3) + (-a*b^2 -
(a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x)) - 2*sqrt(3)*(b^2*c*d + b^2*d^2 + (c*d + d^2)*x^2 - 2*(
b*c*d + b*d^2)*x)*arctan(1/3*sqrt(3)*((b - x)*d^(1/3) - 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/
3))/((b - x)*d^(1/3)))/d^(1/3) - 3*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d)/((a - b)*d*x^2 -
2*(a*b - b^2)*d*x + (a*b^2 - b^3)*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {- a - b c + c x + x}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (- b + x\right ) \left (a - b d + d x - x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)**2)**(1/3)/(a-b*d+(-1+d)*x),x)

[Out]

Integral((-a - b*c + c*x + x)/(((-a + x)*(-b + x)**2)**(1/3)*(-b + x)*(a - b*d + d*x - x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(a-b*d+(-1+d)*x),x, algorithm="giac")

[Out]

integrate(-(b*c - (c + 1)*x + a)/((-(a - x)*(b - x)^2)^(1/3)*(b*d - (d - 1)*x - a)*(b - x)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int -\frac {a+b\,c-x\,\left (c+1\right )}{\left (b-x\right )\,{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (a-b\,d+x\,\left (d-1\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*c - x*(c + 1))/((b - x)*(-(a - x)*(b - x)^2)^(1/3)*(a - b*d + x*(d - 1))),x)

[Out]

-int(-(a + b*c - x*(c + 1))/((b - x)*(-(a - x)*(b - x)^2)^(1/3)*(a - b*d + x*(d - 1))), x)

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