3.2884 \(\int \frac {1}{(c+d x)^3 (a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=219 \[ -\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}+\frac {20 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} d}-\frac {10}{9 a^3 d (c+d x)^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \]

[Out]

-10/9/a^3/d/(d*x+c)^2+1/6/a/d/(d*x+c)^2/(a+b*(d*x+c)^3)^2+4/9/a^2/d/(d*x+c)^2/(a+b*(d*x+c)^3)-20/27*b^(2/3)*ln
(a^(1/3)+b^(1/3)*(d*x+c))/a^(11/3)/d+10/27*b^(2/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(11
/3)/d+20/27*b^(2/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))/a^(1/3)*3^(1/2))/a^(11/3)/d*3^(1/2)

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Rubi [A]  time = 0.18, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {372, 290, 325, 200, 31, 634, 617, 204, 628} \[ -\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}+\frac {20 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} d}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-10/(9*a^3*d*(c + d*x)^2) + 1/(6*a*d*(c + d*x)^2*(a + b*(c + d*x)^3)^2) + 4/(9*a^2*d*(c + d*x)^2*(a + b*(c + d
*x)^3)) + (20*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*d) - (20*
b^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(11/3)*d) + (10*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x
) + b^(2/3)*(c + d*x)^2])/(27*a^(11/3)*d)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{(c+d x)^3 \left (a+b (c+d x)^3\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 a d}\\ &=\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {20 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{9 a^2 d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {(20 b) \operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,c+d x\right )}{9 a^3 d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {(20 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{11/3} d}-\frac {(20 b) \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{27 a^{11/3} d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {\left (10 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{27 a^{11/3} d}-\frac {(10 b) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{9 a^{10/3} d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}-\frac {\left (20 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{9 a^{11/3} d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {20 b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{11/3} d}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 192, normalized size = 0.88 \[ \frac {20 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac {9 a^{5/3} b (c+d x)}{\left (a+b (c+d x)^3\right )^2}-\frac {33 a^{2/3} b (c+d x)}{a+b (c+d x)^3}-\frac {27 a^{2/3}}{(c+d x)^2}-40 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-40 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{54 a^{11/3} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-27*a^(2/3))/(c + d*x)^2 - (9*a^(5/3)*b*(c + d*x))/(a + b*(c + d*x)^3)^2 - (33*a^(2/3)*b*(c + d*x))/(a + b*(
c + d*x)^3) - 40*Sqrt[3]*b^(2/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))] - 40*b^(2/3)*Log[a
^(1/3) + b^(1/3)*(c + d*x)] + 20*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(54*a
^(11/3)*d)

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fricas [B]  time = 1.28, size = 1024, normalized size = 4.68 \[ -\frac {60 \, b^{2} d^{6} x^{6} + 360 \, b^{2} c d^{5} x^{5} + 900 \, b^{2} c^{2} d^{4} x^{4} + 60 \, b^{2} c^{6} + 48 \, {\left (25 \, b^{2} c^{3} + 2 \, a b\right )} d^{3} x^{3} + 96 \, a b c^{3} + 36 \, {\left (25 \, b^{2} c^{4} + 8 \, a b c\right )} d^{2} x^{2} + 72 \, {\left (5 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d x - 40 \, \sqrt {3} {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (a d x + a c\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 20 \, {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + {\left (a b d x + a b c\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 40 \, {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b d x + b c - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 27 \, a^{2}}{54 \, {\left (a^{3} b^{2} d^{9} x^{8} + 8 \, a^{3} b^{2} c d^{8} x^{7} + 28 \, a^{3} b^{2} c^{2} d^{7} x^{6} + 2 \, {\left (28 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{6} x^{5} + 10 \, {\left (7 \, a^{3} b^{2} c^{4} + a^{4} b c\right )} d^{5} x^{4} + 4 \, {\left (14 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{4} x^{3} + {\left (28 \, a^{3} b^{2} c^{6} + 20 \, a^{4} b c^{3} + a^{5}\right )} d^{3} x^{2} + 2 \, {\left (4 \, a^{3} b^{2} c^{7} + 5 \, a^{4} b c^{4} + a^{5} c\right )} d^{2} x + {\left (a^{3} b^{2} c^{8} + 2 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(60*b^2*d^6*x^6 + 360*b^2*c*d^5*x^5 + 900*b^2*c^2*d^4*x^4 + 60*b^2*c^6 + 48*(25*b^2*c^3 + 2*a*b)*d^3*x^3
+ 96*a*b*c^3 + 36*(25*b^2*c^4 + 8*a*b*c)*d^2*x^2 + 72*(5*b^2*c^5 + 4*a*b*c^2)*d*x - 40*sqrt(3)*(b^2*d^8*x^8 +
8*b^2*c*d^7*x^7 + 28*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x^5 + b^2*c^8 + 10*(7*b^2*c^4 + a*b*c)*d^4*x^
4 + 2*a*b*c^5 + 4*(14*b^2*c^5 + 5*a*b*c^2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a^2)*d^2*x^2 + a^2*c^2 + 2*(4*
b^2*c^7 + 5*a*b*c^4 + a^2*c)*d*x)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*(a*d*x + a*c)*(-b^2/a^2)^(2/3) - sqrt
(3)*b)/b) + 20*(b^2*d^8*x^8 + 8*b^2*c*d^7*x^7 + 28*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x^5 + b^2*c^8 +
10*(7*b^2*c^4 + a*b*c)*d^4*x^4 + 2*a*b*c^5 + 4*(14*b^2*c^5 + 5*a*b*c^2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a
^2)*d^2*x^2 + a^2*c^2 + 2*(4*b^2*c^7 + 5*a*b*c^4 + a^2*c)*d*x)*(-b^2/a^2)^(1/3)*log(b^2*d^2*x^2 + 2*b^2*c*d*x
+ b^2*c^2 + a^2*(-b^2/a^2)^(2/3) + (a*b*d*x + a*b*c)*(-b^2/a^2)^(1/3)) - 40*(b^2*d^8*x^8 + 8*b^2*c*d^7*x^7 + 2
8*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x^5 + b^2*c^8 + 10*(7*b^2*c^4 + a*b*c)*d^4*x^4 + 2*a*b*c^5 + 4*(1
4*b^2*c^5 + 5*a*b*c^2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a^2)*d^2*x^2 + a^2*c^2 + 2*(4*b^2*c^7 + 5*a*b*c^4
+ a^2*c)*d*x)*(-b^2/a^2)^(1/3)*log(b*d*x + b*c - a*(-b^2/a^2)^(1/3)) + 27*a^2)/(a^3*b^2*d^9*x^8 + 8*a^3*b^2*c*
d^8*x^7 + 28*a^3*b^2*c^2*d^7*x^6 + 2*(28*a^3*b^2*c^3 + a^4*b)*d^6*x^5 + 10*(7*a^3*b^2*c^4 + a^4*b*c)*d^5*x^4 +
4*(14*a^3*b^2*c^5 + 5*a^4*b*c^2)*d^4*x^3 + (28*a^3*b^2*c^6 + 20*a^4*b*c^3 + a^5)*d^3*x^2 + 2*(4*a^3*b^2*c^7 +
5*a^4*b*c^4 + a^5*c)*d^2*x + (a^3*b^2*c^8 + 2*a^4*b*c^5 + a^5*c^2)*d)

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giac [B]  time = 0.27, size = 365, normalized size = 1.67 \[ \frac {10 \, {\left (2 \, \sqrt {3} \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | -b d x - b c + \left (-a b^{2}\right )^{\frac {1}{3}} \right |}\right )\right )}}{27 \, a^{3}} - \frac {20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 400 \, b^{2} c^{3} d^{3} x^{3} + 300 \, b^{2} c^{4} d^{2} x^{2} + 120 \, b^{2} c^{5} d x + 20 \, b^{2} c^{6} + 32 \, a b d^{3} x^{3} + 96 \, a b c d^{2} x^{2} + 96 \, a b c^{2} d x + 32 \, a b c^{3} + 9 \, a^{2}}{18 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a d x + a c\right )}^{2} a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/27*(2*sqrt(3)*(-b^2/(a^2*d^3))^(1/3)*arctan(-(b*d*x + b*c - (-a*b^2)^(1/3))/(sqrt(3)*b*d*x + sqrt(3)*b*c +
sqrt(3)*(-a*b^2)^(1/3))) - (-b^2/(a^2*d^3))^(1/3)*log(4*(sqrt(3)*b*d*x + sqrt(3)*b*c + sqrt(3)*(-a*b^2)^(1/3))
^2 + 4*(b*d*x + b*c - (-a*b^2)^(1/3))^2) + 2*(-b^2/(a^2*d^3))^(1/3)*log(abs(-b*d*x - b*c + (-a*b^2)^(1/3))))/a
^3 - 1/18*(20*b^2*d^6*x^6 + 120*b^2*c*d^5*x^5 + 300*b^2*c^2*d^4*x^4 + 400*b^2*c^3*d^3*x^3 + 300*b^2*c^4*d^2*x^
2 + 120*b^2*c^5*d*x + 20*b^2*c^6 + 32*a*b*d^3*x^3 + 96*a*b*c*d^2*x^2 + 96*a*b*c^2*d*x + 32*a*b*c^3 + 9*a^2)/((
b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4 + a*d*x + a*c)^2*a^3*d)

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maple [C]  time = 0.03, size = 419, normalized size = 1.91 \[ -\frac {11 b^{2} d^{3} x^{4}}{18 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3}}-\frac {22 b^{2} c \,d^{2} x^{3}}{9 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3}}-\frac {11 b^{2} c^{2} d \,x^{2}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3}}-\frac {22 b^{2} c^{3} x}{9 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3}}-\frac {11 b^{2} c^{4}}{18 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3} d}-\frac {7 b x}{9 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2}}-\frac {7 b c}{9 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2} d}-\frac {20 \ln \left (-\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+x \right )}{27 a^{3} d \left (d^{2} \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )^{2}+2 c d \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+c^{2}\right )}-\frac {1}{2 \left (d x +c \right )^{2} a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a^3/d/(d*x+c)^2-11/18*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*d^3*x^4-22/9*b^2/a^3/(b*d^3
*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c*d^2*x^3-11/3*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)
^2*c^2*d*x^2-22/9*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*x*c^3-7/9*b/a^2/(b*d^3*x^3+3*b*c*d^2
*x^2+3*b*c^2*d*x+b*c^3+a)^2*x-11/18*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^4/d-7/9*b/a^2/(b
*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c/d-20/27/a^3/d*sum(1/(_R^2*d^2+2*_R*c*d+c^2)*ln(-_R+x),_R=RootO
f(_Z^3*b*d^3+3*_Z^2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{6} + 16 \, {\left (25 \, b^{2} c^{3} + 2 \, a b\right )} d^{3} x^{3} + 32 \, a b c^{3} + 12 \, {\left (25 \, b^{2} c^{4} + 8 \, a b c\right )} d^{2} x^{2} + 24 \, {\left (5 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d x + 9 \, a^{2}}{18 \, {\left (a^{3} b^{2} d^{9} x^{8} + 8 \, a^{3} b^{2} c d^{8} x^{7} + 28 \, a^{3} b^{2} c^{2} d^{7} x^{6} + 2 \, {\left (28 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{6} x^{5} + 10 \, {\left (7 \, a^{3} b^{2} c^{4} + a^{4} b c\right )} d^{5} x^{4} + 4 \, {\left (14 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{4} x^{3} + {\left (28 \, a^{3} b^{2} c^{6} + 20 \, a^{4} b c^{3} + a^{5}\right )} d^{3} x^{2} + 2 \, {\left (4 \, a^{3} b^{2} c^{7} + 5 \, a^{4} b c^{4} + a^{5} c\right )} d^{2} x + {\left (a^{3} b^{2} c^{8} + 2 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d\right )}} - \frac {\frac {10}{3} \, b {\left (\frac {2 \, \sqrt {3} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}}\right )}{d} - \frac {\left (\frac {1}{a^{2} b}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right )}{d} + \frac {2 \, \left (\frac {1}{a^{2} b}\right )^{\frac {1}{3}} \log \left ({\left | b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}} \right |}\right )}{d}\right )}}{9 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(20*b^2*d^6*x^6 + 120*b^2*c*d^5*x^5 + 300*b^2*c^2*d^4*x^4 + 20*b^2*c^6 + 16*(25*b^2*c^3 + 2*a*b)*d^3*x^3
+ 32*a*b*c^3 + 12*(25*b^2*c^4 + 8*a*b*c)*d^2*x^2 + 24*(5*b^2*c^5 + 4*a*b*c^2)*d*x + 9*a^2)/(a^3*b^2*d^9*x^8 +
8*a^3*b^2*c*d^8*x^7 + 28*a^3*b^2*c^2*d^7*x^6 + 2*(28*a^3*b^2*c^3 + a^4*b)*d^6*x^5 + 10*(7*a^3*b^2*c^4 + a^4*b
*c)*d^5*x^4 + 4*(14*a^3*b^2*c^5 + 5*a^4*b*c^2)*d^4*x^3 + (28*a^3*b^2*c^6 + 20*a^4*b*c^3 + a^5)*d^3*x^2 + 2*(4*
a^3*b^2*c^7 + 5*a^4*b*c^4 + a^5*c)*d^2*x + (a^3*b^2*c^8 + 2*a^4*b*c^5 + a^5*c^2)*d) - 20/9*b*integrate(1/(b*d^
3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/a^3

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mupad [B]  time = 2.69, size = 520, normalized size = 2.37 \[ \frac {20\,b^{2/3}\,\ln \left (a^6\,b^{1/3}\,c-{\left (-a\right )}^{19/3}+a^6\,b^{1/3}\,d\,x\right )}{27\,{\left (-a\right )}^{11/3}\,d}-\frac {\frac {9\,a^2+32\,a\,b\,c^3+20\,b^2\,c^6}{18\,a^3\,d}+\frac {2\,x^2\,\left (25\,d\,b^2\,c^4+8\,a\,d\,b\,c\right )}{3\,a^3}+\frac {4\,x\,\left (5\,b^2\,c^5+4\,a\,b\,c^2\right )}{3\,a^3}+\frac {8\,x^3\,\left (25\,b^2\,c^3\,d^2+2\,a\,b\,d^2\right )}{9\,a^3}+\frac {10\,b^2\,d^5\,x^6}{9\,a^3}+\frac {50\,b^2\,c^2\,d^3\,x^4}{3\,a^3}+\frac {20\,b^2\,c\,d^4\,x^5}{3\,a^3}}{x^5\,\left (56\,b^2\,c^3\,d^5+2\,a\,b\,d^5\right )+x^4\,\left (70\,b^2\,c^4\,d^4+10\,a\,b\,c\,d^4\right )+x\,\left (2\,d\,a^2\,c+10\,d\,a\,b\,c^4+8\,d\,b^2\,c^7\right )+x^2\,\left (a^2\,d^2+20\,a\,b\,c^3\,d^2+28\,b^2\,c^6\,d^2\right )+a^2\,c^2+b^2\,c^8+x^3\,\left (56\,b^2\,c^5\,d^3+20\,a\,b\,c^2\,d^3\right )+b^2\,d^8\,x^8+2\,a\,b\,c^5+8\,b^2\,c\,d^7\,x^7+28\,b^2\,c^2\,d^6\,x^6}+\frac {20\,b^{2/3}\,\ln \left (4860\,a^6\,b^3\,c\,d^5+4860\,a^6\,b^3\,d^6\,x-4860\,{\left (-a\right )}^{19/3}\,b^{8/3}\,d^5\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,{\left (-a\right )}^{11/3}\,d}-\frac {20\,b^{2/3}\,\ln \left (4860\,a^6\,b^3\,c\,d^5+4860\,a^6\,b^3\,d^6\,x+4860\,{\left (-a\right )}^{19/3}\,b^{8/3}\,d^5\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,{\left (-a\right )}^{11/3}\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(20*b^(2/3)*log(a^6*b^(1/3)*c - (-a)^(19/3) + a^6*b^(1/3)*d*x))/(27*(-a)^(11/3)*d) - ((9*a^2 + 20*b^2*c^6 + 32
*a*b*c^3)/(18*a^3*d) + (2*x^2*(25*b^2*c^4*d + 8*a*b*c*d))/(3*a^3) + (4*x*(5*b^2*c^5 + 4*a*b*c^2))/(3*a^3) + (8
*x^3*(25*b^2*c^3*d^2 + 2*a*b*d^2))/(9*a^3) + (10*b^2*d^5*x^6)/(9*a^3) + (50*b^2*c^2*d^3*x^4)/(3*a^3) + (20*b^2
*c*d^4*x^5)/(3*a^3))/(x^5*(56*b^2*c^3*d^5 + 2*a*b*d^5) + x^4*(70*b^2*c^4*d^4 + 10*a*b*c*d^4) + x*(8*b^2*c^7*d
+ 2*a^2*c*d + 10*a*b*c^4*d) + x^2*(a^2*d^2 + 28*b^2*c^6*d^2 + 20*a*b*c^3*d^2) + a^2*c^2 + b^2*c^8 + x^3*(56*b^
2*c^5*d^3 + 20*a*b*c^2*d^3) + b^2*d^8*x^8 + 2*a*b*c^5 + 8*b^2*c*d^7*x^7 + 28*b^2*c^2*d^6*x^6) + (20*b^(2/3)*lo
g(4860*a^6*b^3*c*d^5 + 4860*a^6*b^3*d^6*x - 4860*(-a)^(19/3)*b^(8/3)*d^5*((3^(1/2)*1i)/2 - 1/2))*((3^(1/2)*1i)
/2 - 1/2))/(27*(-a)^(11/3)*d) - (20*b^(2/3)*log(4860*a^6*b^3*c*d^5 + 4860*a^6*b^3*d^6*x + 4860*(-a)^(19/3)*b^(
8/3)*d^5*((3^(1/2)*1i)/2 + 1/2))*((3^(1/2)*1i)/2 + 1/2))/(27*(-a)^(11/3)*d)

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sympy [B]  time = 6.63, size = 435, normalized size = 1.99 \[ \frac {- 9 a^{2} - 32 a b c^{3} - 20 b^{2} c^{6} - 300 b^{2} c^{2} d^{4} x^{4} - 120 b^{2} c d^{5} x^{5} - 20 b^{2} d^{6} x^{6} + x^{3} \left (- 32 a b d^{3} - 400 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 96 a b c d^{2} - 300 b^{2} c^{4} d^{2}\right ) + x \left (- 96 a b c^{2} d - 120 b^{2} c^{5} d\right )}{18 a^{5} c^{2} d + 36 a^{4} b c^{5} d + 18 a^{3} b^{2} c^{8} d + 504 a^{3} b^{2} c^{2} d^{7} x^{6} + 144 a^{3} b^{2} c d^{8} x^{7} + 18 a^{3} b^{2} d^{9} x^{8} + x^{5} \left (36 a^{4} b d^{6} + 1008 a^{3} b^{2} c^{3} d^{6}\right ) + x^{4} \left (180 a^{4} b c d^{5} + 1260 a^{3} b^{2} c^{4} d^{5}\right ) + x^{3} \left (360 a^{4} b c^{2} d^{4} + 1008 a^{3} b^{2} c^{5} d^{4}\right ) + x^{2} \left (18 a^{5} d^{3} + 360 a^{4} b c^{3} d^{3} + 504 a^{3} b^{2} c^{6} d^{3}\right ) + x \left (36 a^{5} c d^{2} + 180 a^{4} b c^{4} d^{2} + 144 a^{3} b^{2} c^{7} d^{2}\right )} + \frac {\operatorname {RootSum} {\left (19683 t^{3} a^{11} + 8000 b^{2}, \left (t \mapsto t \log {\left (x + \frac {- 27 t a^{4} + 20 b c}{20 b d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-9*a**2 - 32*a*b*c**3 - 20*b**2*c**6 - 300*b**2*c**2*d**4*x**4 - 120*b**2*c*d**5*x**5 - 20*b**2*d**6*x**6 + x
**3*(-32*a*b*d**3 - 400*b**2*c**3*d**3) + x**2*(-96*a*b*c*d**2 - 300*b**2*c**4*d**2) + x*(-96*a*b*c**2*d - 120
*b**2*c**5*d))/(18*a**5*c**2*d + 36*a**4*b*c**5*d + 18*a**3*b**2*c**8*d + 504*a**3*b**2*c**2*d**7*x**6 + 144*a
**3*b**2*c*d**8*x**7 + 18*a**3*b**2*d**9*x**8 + x**5*(36*a**4*b*d**6 + 1008*a**3*b**2*c**3*d**6) + x**4*(180*a
**4*b*c*d**5 + 1260*a**3*b**2*c**4*d**5) + x**3*(360*a**4*b*c**2*d**4 + 1008*a**3*b**2*c**5*d**4) + x**2*(18*a
**5*d**3 + 360*a**4*b*c**3*d**3 + 504*a**3*b**2*c**6*d**3) + x*(36*a**5*c*d**2 + 180*a**4*b*c**4*d**2 + 144*a*
*3*b**2*c**7*d**2)) + RootSum(19683*_t**3*a**11 + 8000*b**2, Lambda(_t, _t*log(x + (-27*_t*a**4 + 20*b*c)/(20*
b*d))))/d

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