3.24.51 \(\int \frac {1}{(c e+d e x)^2 (a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=237 \[ -\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d e^2}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d e^2}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} d e^2}-\frac {14}{9 a^3 d e^2 (c+d x)}+\frac {7}{18 a^2 d e^2 (c+d x) \left (a+b (c+d x)^3\right )}+\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2} \]

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Rubi [A]  time = 0.18, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {372, 290, 325, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d e^2}+\frac {7}{18 a^2 d e^2 (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d e^2}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} d e^2}-\frac {14}{9 a^3 d e^2 (c+d x)}+\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-14/(9*a^3*d*e^2*(c + d*x)) + 1/(6*a*d*e^2*(c + d*x)*(a + b*(c + d*x)^3)^2) + 7/(18*a^2*d*e^2*(c + d*x)*(a + b
*(c + d*x)^3)) + (14*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*d*
e^2) + (14*b^(1/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(10/3)*d*e^2) - (7*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^
(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(27*a^(10/3)*d*e^2)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, 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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 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 e+d e x)^2 \left (a+b (c+d x)^3\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d e^2}\\ &=\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{6 a d e^2}\\ &=\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d e^2 (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{9 a^2 d e^2}\\ &=-\frac {14}{9 a^3 d e^2 (c+d x)}+\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d e^2 (c+d x) \left (a+b (c+d x)^3\right )}-\frac {(14 b) \operatorname {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{9 a^3 d e^2}\\ &=-\frac {14}{9 a^3 d e^2 (c+d x)}+\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d e^2 (c+d x) \left (a+b (c+d x)^3\right )}+\frac {\left (14 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{10/3} d e^2}-\frac {\left (14 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\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^{10/3} d e^2}\\ &=-\frac {14}{9 a^3 d e^2 (c+d x)}+\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d e^2 (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d e^2}-\frac {\left (7 \sqrt [3]{b}\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^{10/3} d e^2}-\frac {\left (7 b^{2/3}\right ) \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^3 d e^2}\\ &=-\frac {14}{9 a^3 d e^2 (c+d x)}+\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d e^2 (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d e^2}-\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d e^2}-\frac {\left (14 \sqrt [3]{b}\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^{10/3} d e^2}\\ &=-\frac {14}{9 a^3 d e^2 (c+d x)}+\frac {1}{6 a d e^2 (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d e^2 (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{10/3} d e^2}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d e^2}-\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d e^2}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 199, normalized size = 0.84 \begin {gather*} \frac {-14 \sqrt [3]{b} \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^{4/3} b (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac {30 \sqrt [3]{a} b (c+d x)^2}{a+b (c+d x)^3}+28 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-28 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {54 \sqrt [3]{a}}{c+d x}}{54 a^{10/3} d e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-54*a^(1/3))/(c + d*x) - (9*a^(4/3)*b*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 - (30*a^(1/3)*b*(c + d*x)^2)/(a + b
*(c + d*x)^3) - 28*Sqrt[3]*b^(1/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))] + 28*b^(1/3)*Log
[a^(1/3) + b^(1/3)*(c + d*x)] - 14*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(54
*a^(10/3)*d*e^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.08, size = 879, normalized size = 3.71 \begin {gather*} -\frac {84 \, b^{2} d^{6} x^{6} + 504 \, b^{2} c d^{5} x^{5} + 1260 \, b^{2} c^{2} d^{4} x^{4} + 84 \, b^{2} c^{6} + 21 \, {\left (80 \, b^{2} c^{3} + 7 \, a b\right )} d^{3} x^{3} + 147 \, a b c^{3} + 63 \, {\left (20 \, b^{2} c^{4} + 7 \, a b c\right )} d^{2} x^{2} + 63 \, {\left (8 \, b^{2} c^{5} + 7 \, a b c^{2}\right )} d x + 28 \, \sqrt {3} {\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + b^{2} c^{7} + {\left (35 \, b^{2} c^{3} + 2 \, a b\right )} d^{4} x^{4} + {\left (35 \, b^{2} c^{4} + 8 \, a b c\right )} d^{3} x^{3} + 2 \, a b c^{4} + 3 \, {\left (7 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d^{2} x^{2} + a^{2} c + {\left (7 \, b^{2} c^{6} + 8 \, a b c^{3} + a^{2}\right )} d x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 14 \, {\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + b^{2} c^{7} + {\left (35 \, b^{2} c^{3} + 2 \, a b\right )} d^{4} x^{4} + {\left (35 \, b^{2} c^{4} + 8 \, a b c\right )} d^{3} x^{3} + 2 \, a b c^{4} + 3 \, {\left (7 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d^{2} x^{2} + a^{2} c + {\left (7 \, b^{2} c^{6} + 8 \, a b c^{3} + a^{2}\right )} d x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - {\left (a d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 28 \, {\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + b^{2} c^{7} + {\left (35 \, b^{2} c^{3} + 2 \, a b\right )} d^{4} x^{4} + {\left (35 \, b^{2} c^{4} + 8 \, a b c\right )} d^{3} x^{3} + 2 \, a b c^{4} + 3 \, {\left (7 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d^{2} x^{2} + a^{2} c + {\left (7 \, b^{2} c^{6} + 8 \, a b c^{3} + a^{2}\right )} d x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d x + b c + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 54 \, a^{2}}{54 \, {\left (a^{3} b^{2} d^{8} e^{2} x^{7} + 7 \, a^{3} b^{2} c d^{7} e^{2} x^{6} + 21 \, a^{3} b^{2} c^{2} d^{6} e^{2} x^{5} + {\left (35 \, a^{3} b^{2} c^{3} + 2 \, a^{4} b\right )} d^{5} e^{2} x^{4} + {\left (35 \, a^{3} b^{2} c^{4} + 8 \, a^{4} b c\right )} d^{4} e^{2} x^{3} + 3 \, {\left (7 \, a^{3} b^{2} c^{5} + 4 \, a^{4} b c^{2}\right )} d^{3} e^{2} x^{2} + {\left (7 \, a^{3} b^{2} c^{6} + 8 \, a^{4} b c^{3} + a^{5}\right )} d^{2} e^{2} x + {\left (a^{3} b^{2} c^{7} + 2 \, a^{4} b c^{4} + a^{5} c\right )} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(84*b^2*d^6*x^6 + 504*b^2*c*d^5*x^5 + 1260*b^2*c^2*d^4*x^4 + 84*b^2*c^6 + 21*(80*b^2*c^3 + 7*a*b)*d^3*x^
3 + 147*a*b*c^3 + 63*(20*b^2*c^4 + 7*a*b*c)*d^2*x^2 + 63*(8*b^2*c^5 + 7*a*b*c^2)*d*x + 28*sqrt(3)*(b^2*d^7*x^7
+ 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^5 + b^2*c^7 + (35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^2*c^4 + 8*a*b*c)*d^3*
x^3 + 2*a*b*c^4 + 3*(7*b^2*c^5 + 4*a*b*c^2)*d^2*x^2 + a^2*c + (7*b^2*c^6 + 8*a*b*c^3 + a^2)*d*x)*(b/a)^(1/3)*a
rctan(2/3*sqrt(3)*(d*x + c)*(b/a)^(1/3) - 1/3*sqrt(3)) + 14*(b^2*d^7*x^7 + 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^
5 + b^2*c^7 + (35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^2*c^4 + 8*a*b*c)*d^3*x^3 + 2*a*b*c^4 + 3*(7*b^2*c^5 + 4*a*b
*c^2)*d^2*x^2 + a^2*c + (7*b^2*c^6 + 8*a*b*c^3 + a^2)*d*x)*(b/a)^(1/3)*log(b*d^2*x^2 + 2*b*c*d*x + b*c^2 - (a*
d*x + a*c)*(b/a)^(2/3) + a*(b/a)^(1/3)) - 28*(b^2*d^7*x^7 + 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^5 + b^2*c^7 + (
35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^2*c^4 + 8*a*b*c)*d^3*x^3 + 2*a*b*c^4 + 3*(7*b^2*c^5 + 4*a*b*c^2)*d^2*x^2 +
a^2*c + (7*b^2*c^6 + 8*a*b*c^3 + a^2)*d*x)*(b/a)^(1/3)*log(b*d*x + b*c + a*(b/a)^(2/3)) + 54*a^2)/(a^3*b^2*d^
8*e^2*x^7 + 7*a^3*b^2*c*d^7*e^2*x^6 + 21*a^3*b^2*c^2*d^6*e^2*x^5 + (35*a^3*b^2*c^3 + 2*a^4*b)*d^5*e^2*x^4 + (3
5*a^3*b^2*c^4 + 8*a^4*b*c)*d^4*e^2*x^3 + 3*(7*a^3*b^2*c^5 + 4*a^4*b*c^2)*d^3*e^2*x^2 + (7*a^3*b^2*c^6 + 8*a^4*
b*c^3 + a^5)*d^2*e^2*x + (a^3*b^2*c^7 + 2*a^4*b*c^4 + a^5*c)*d*e^2)

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giac [A]  time = 0.27, size = 295, normalized size = 1.24 \begin {gather*} \frac {14 \, \left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (-2\right )} \log \left ({\left | -\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (-2\right )} - \frac {e^{\left (-1\right )}}{{\left (d x e + c e\right )} d} \right |}\right )}{27 \, a^{3}} - \frac {14 \, \sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (-2\right )} - \frac {2 \, e^{\left (-1\right )}}{{\left (d x e + c e\right )} d}\right )} e^{2}}{3 \, \left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}\right ) e^{\left (-2\right )}}{27 \, a^{4} d} - \frac {7 \, \left (a^{2} b\right )^{\frac {1}{3}} e^{\left (-2\right )} \log \left (\left (\frac {b}{a d^{3}}\right )^{\frac {2}{3}} e^{\left (-4\right )} - \frac {\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} e^{\left (-3\right )}}{{\left (d x e + c e\right )} d} + \frac {e^{\left (-2\right )}}{{\left (d x e + c e\right )}^{2} d^{2}}\right )}{27 \, a^{4} d} - \frac {\frac {10 \, b^{2} e^{\left (-1\right )}}{{\left (d x e + c e\right )} d} + \frac {13 \, a b e^{2}}{{\left (d x e + c e\right )}^{4} d}}{18 \, a^{3} {\left (b + \frac {a e^{3}}{{\left (d x e + c e\right )}^{3}}\right )}^{2}} - \frac {e^{\left (-1\right )}}{{\left (d x e + c e\right )} a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14/27*(b/(a*d^3))^(1/3)*e^(-2)*log(abs(-(b/(a*d^3))^(1/3)*e^(-2) - e^(-1)/((d*x*e + c*e)*d)))/a^3 - 14/27*sqrt
(3)*(a^2*b)^(1/3)*arctan(1/3*sqrt(3)*((b/(a*d^3))^(1/3)*e^(-2) - 2*e^(-1)/((d*x*e + c*e)*d))*e^2/(b/(a*d^3))^(
1/3))*e^(-2)/(a^4*d) - 7/27*(a^2*b)^(1/3)*e^(-2)*log((b/(a*d^3))^(2/3)*e^(-4) - (b/(a*d^3))^(1/3)*e^(-3)/((d*x
*e + c*e)*d) + e^(-2)/((d*x*e + c*e)^2*d^2))/(a^4*d) - 1/18*(10*b^2*e^(-1)/((d*x*e + c*e)*d) + 13*a*b*e^2/((d*
x*e + c*e)^4*d))/(a^3*(b + a*e^3/(d*x*e + c*e)^3)^2) - e^(-1)/((d*x*e + c*e)*a^3*d)

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maple [C]  time = 0.03, size = 557, normalized size = 2.35 \begin {gather*} -\frac {5 b^{2} d^{4} x^{5}}{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} e^{2}}-\frac {25 b^{2} c \,d^{3} x^{4}}{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} e^{2}}-\frac {50 b^{2} c^{2} 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} e^{2}}-\frac {50 b^{2} c^{3} d \,x^{2}}{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} e^{2}}-\frac {25 b^{2} c^{4} 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} e^{2}}-\frac {5 b^{2} c^{5}}{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} d \,e^{2}}-\frac {13 b d \,x^{2}}{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^{2} e^{2}}-\frac {13 b c 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} e^{2}}-\frac {13 b \,c^{2}}{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^{2} d \,e^{2}}-\frac {14 \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 ) d +c \right ) \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 \,e^{2} \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}{\left (d x +c \right ) a^{3} d \,e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^3/d/e^2/(d*x+c)-5/9/e^2*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^4*x^5-25/9/e^2*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^3*x^4-50/9/e^2*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^2*x^3-50/9/e^2*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^2*c^3*d-13/18/e^2*
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^2*d-25/9/e^2*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^4-13/9/e^2*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*x-5/9/e^2*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^5/d-13/18/e^2*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^2/d-14/27/e^2/a^3/d*sum((_R*d+c)/(_R^2*d^2+2*_R*c*d+c^2)*ln(-_R+x),_R=RootOf(_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 \begin {gather*} -\frac {28 \, b^{2} d^{6} x^{6} + 168 \, b^{2} c d^{5} x^{5} + 420 \, b^{2} c^{2} d^{4} x^{4} + 28 \, b^{2} c^{6} + 7 \, {\left (80 \, b^{2} c^{3} + 7 \, a b\right )} d^{3} x^{3} + 49 \, a b c^{3} + 21 \, {\left (20 \, b^{2} c^{4} + 7 \, a b c\right )} d^{2} x^{2} + 21 \, {\left (8 \, b^{2} c^{5} + 7 \, a b c^{2}\right )} d x + 18 \, a^{2}}{18 \, {\left (a^{3} b^{2} d^{8} e^{2} x^{7} + 7 \, a^{3} b^{2} c d^{7} e^{2} x^{6} + 21 \, a^{3} b^{2} c^{2} d^{6} e^{2} x^{5} + {\left (35 \, a^{3} b^{2} c^{3} + 2 \, a^{4} b\right )} d^{5} e^{2} x^{4} + {\left (35 \, a^{3} b^{2} c^{4} + 8 \, a^{4} b c\right )} d^{4} e^{2} x^{3} + 3 \, {\left (7 \, a^{3} b^{2} c^{5} + 4 \, a^{4} b c^{2}\right )} d^{3} e^{2} x^{2} + {\left (7 \, a^{3} b^{2} c^{6} + 8 \, a^{4} b c^{3} + a^{5}\right )} d^{2} e^{2} x + {\left (a^{3} b^{2} c^{7} + 2 \, a^{4} b c^{4} + a^{5} c\right )} d e^{2}\right )}} - \frac {-\frac {7}{3} \, {\left (2 \, \sqrt {3} \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )\right )} b}{9 \, a^{3} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(28*b^2*d^6*x^6 + 168*b^2*c*d^5*x^5 + 420*b^2*c^2*d^4*x^4 + 28*b^2*c^6 + 7*(80*b^2*c^3 + 7*a*b)*d^3*x^3
+ 49*a*b*c^3 + 21*(20*b^2*c^4 + 7*a*b*c)*d^2*x^2 + 21*(8*b^2*c^5 + 7*a*b*c^2)*d*x + 18*a^2)/(a^3*b^2*d^8*e^2*x
^7 + 7*a^3*b^2*c*d^7*e^2*x^6 + 21*a^3*b^2*c^2*d^6*e^2*x^5 + (35*a^3*b^2*c^3 + 2*a^4*b)*d^5*e^2*x^4 + (35*a^3*b
^2*c^4 + 8*a^4*b*c)*d^4*e^2*x^3 + 3*(7*a^3*b^2*c^5 + 4*a^4*b*c^2)*d^3*e^2*x^2 + (7*a^3*b^2*c^6 + 8*a^4*b*c^3 +
a^5)*d^2*e^2*x + (a^3*b^2*c^7 + 2*a^4*b*c^4 + a^5*c)*d*e^2) - 14/9*b*integrate((d*x + c)/(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*e^2)

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mupad [B]  time = 2.60, size = 485, normalized size = 2.05 \begin {gather*} \frac {14\,b^{1/3}\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{27\,a^{10/3}\,d\,e^2}-\frac {\frac {18\,a^2+49\,a\,b\,c^3+28\,b^2\,c^6}{18\,a^3\,d}+\frac {7\,x^2\,\left (20\,d\,b^2\,c^4+7\,a\,d\,b\,c\right )}{6\,a^3}+\frac {7\,x\,\left (8\,b^2\,c^5+7\,a\,b\,c^2\right )}{6\,a^3}+\frac {7\,x^3\,\left (80\,b^2\,c^3\,d^2+7\,a\,b\,d^2\right )}{18\,a^3}+\frac {14\,b^2\,d^5\,x^6}{9\,a^3}+\frac {70\,b^2\,c^2\,d^3\,x^4}{3\,a^3}+\frac {28\,b^2\,c\,d^4\,x^5}{3\,a^3}}{x^3\,\left (35\,b^2\,c^4\,d^3\,e^2+8\,a\,b\,c\,d^3\,e^2\right )+x^2\,\left (21\,b^2\,c^5\,d^2\,e^2+12\,a\,b\,c^2\,d^2\,e^2\right )+x\,\left (d\,a^2\,e^2+8\,d\,a\,b\,c^3\,e^2+7\,d\,b^2\,c^6\,e^2\right )+x^4\,\left (35\,b^2\,c^3\,d^4\,e^2+2\,a\,b\,d^4\,e^2\right )+a^2\,c\,e^2+b^2\,c^7\,e^2+b^2\,d^7\,e^2\,x^7+2\,a\,b\,c^4\,e^2+21\,b^2\,c^2\,d^5\,e^2\,x^5+7\,b^2\,c\,d^6\,e^2\,x^6}+\frac {14\,b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,a^{10/3}\,d\,e^2}-\frac {14\,b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,a^{10/3}\,d\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(14*b^(1/3)*log(b^(1/3)*c + a^(1/3) + b^(1/3)*d*x))/(27*a^(10/3)*d*e^2) - ((18*a^2 + 28*b^2*c^6 + 49*a*b*c^3)/
(18*a^3*d) + (7*x^2*(20*b^2*c^4*d + 7*a*b*c*d))/(6*a^3) + (7*x*(8*b^2*c^5 + 7*a*b*c^2))/(6*a^3) + (7*x^3*(80*b
^2*c^3*d^2 + 7*a*b*d^2))/(18*a^3) + (14*b^2*d^5*x^6)/(9*a^3) + (70*b^2*c^2*d^3*x^4)/(3*a^3) + (28*b^2*c*d^4*x^
5)/(3*a^3))/(x^3*(35*b^2*c^4*d^3*e^2 + 8*a*b*c*d^3*e^2) + x^2*(21*b^2*c^5*d^2*e^2 + 12*a*b*c^2*d^2*e^2) + x*(a
^2*d*e^2 + 7*b^2*c^6*d*e^2 + 8*a*b*c^3*d*e^2) + x^4*(35*b^2*c^3*d^4*e^2 + 2*a*b*d^4*e^2) + a^2*c*e^2 + b^2*c^7
*e^2 + b^2*d^7*e^2*x^7 + 2*a*b*c^4*e^2 + 21*b^2*c^2*d^5*e^2*x^5 + 7*b^2*c*d^6*e^2*x^6) + (14*b^(1/3)*log(2*b^(
1/3)*c - 3^(1/2)*a^(1/3)*1i - a^(1/3) + 2*b^(1/3)*d*x)*((3^(1/2)*1i)/2 - 1/2))/(27*a^(10/3)*d*e^2) - (14*b^(1/
3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*c - a^(1/3) + 2*b^(1/3)*d*x)*((3^(1/2)*1i)/2 + 1/2))/(27*a^(10/3)*d*e^2)

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sympy [B]  time = 6.13, size = 445, normalized size = 1.88 \begin {gather*} \frac {- 18 a^{2} - 49 a b c^{3} - 28 b^{2} c^{6} - 420 b^{2} c^{2} d^{4} x^{4} - 168 b^{2} c d^{5} x^{5} - 28 b^{2} d^{6} x^{6} + x^{3} \left (- 49 a b d^{3} - 560 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 147 a b c d^{2} - 420 b^{2} c^{4} d^{2}\right ) + x \left (- 147 a b c^{2} d - 168 b^{2} c^{5} d\right )}{18 a^{5} c d e^{2} + 36 a^{4} b c^{4} d e^{2} + 18 a^{3} b^{2} c^{7} d e^{2} + 378 a^{3} b^{2} c^{2} d^{6} e^{2} x^{5} + 126 a^{3} b^{2} c d^{7} e^{2} x^{6} + 18 a^{3} b^{2} d^{8} e^{2} x^{7} + x^{4} \left (36 a^{4} b d^{5} e^{2} + 630 a^{3} b^{2} c^{3} d^{5} e^{2}\right ) + x^{3} \left (144 a^{4} b c d^{4} e^{2} + 630 a^{3} b^{2} c^{4} d^{4} e^{2}\right ) + x^{2} \left (216 a^{4} b c^{2} d^{3} e^{2} + 378 a^{3} b^{2} c^{5} d^{3} e^{2}\right ) + x \left (18 a^{5} d^{2} e^{2} + 144 a^{4} b c^{3} d^{2} e^{2} + 126 a^{3} b^{2} c^{6} d^{2} e^{2}\right )} + \frac {\operatorname {RootSum} {\left (19683 t^{3} a^{10} - 2744 b, \left (t \mapsto t \log {\left (x + \frac {729 t^{2} a^{7} + 196 b c}{196 b d} \right )} \right )\right )}}{d e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-18*a**2 - 49*a*b*c**3 - 28*b**2*c**6 - 420*b**2*c**2*d**4*x**4 - 168*b**2*c*d**5*x**5 - 28*b**2*d**6*x**6 +
x**3*(-49*a*b*d**3 - 560*b**2*c**3*d**3) + x**2*(-147*a*b*c*d**2 - 420*b**2*c**4*d**2) + x*(-147*a*b*c**2*d -
168*b**2*c**5*d))/(18*a**5*c*d*e**2 + 36*a**4*b*c**4*d*e**2 + 18*a**3*b**2*c**7*d*e**2 + 378*a**3*b**2*c**2*d*
*6*e**2*x**5 + 126*a**3*b**2*c*d**7*e**2*x**6 + 18*a**3*b**2*d**8*e**2*x**7 + x**4*(36*a**4*b*d**5*e**2 + 630*
a**3*b**2*c**3*d**5*e**2) + x**3*(144*a**4*b*c*d**4*e**2 + 630*a**3*b**2*c**4*d**4*e**2) + x**2*(216*a**4*b*c*
*2*d**3*e**2 + 378*a**3*b**2*c**5*d**3*e**2) + x*(18*a**5*d**2*e**2 + 144*a**4*b*c**3*d**2*e**2 + 126*a**3*b**
2*c**6*d**2*e**2)) + RootSum(19683*_t**3*a**10 - 2744*b, Lambda(_t, _t*log(x + (729*_t**2*a**7 + 196*b*c)/(196
*b*d))))/(d*e**2)

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