3.24.11 \(\int \frac {1}{(c+d x)^4 (a+b (c+d x)^3)} \, dx\)

Optimal. Leaf size=56 \[ -\frac {b \log (c+d x)}{a^2 d}+\frac {b \log \left (a+b (c+d x)^3\right )}{3 a^2 d}-\frac {1}{3 a d (c+d x)^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {372, 266, 44} \begin {gather*} -\frac {b \log (c+d x)}{a^2 d}+\frac {b \log \left (a+b (c+d x)^3\right )}{3 a^2 d}-\frac {1}{3 a d (c+d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^4*(a + b*(c + d*x)^3)),x]

[Out]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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]

Rubi steps

\begin {align*} \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d}\\ &=-\frac {1}{3 a d (c+d x)^3}-\frac {b \log (c+d x)}{a^2 d}+\frac {b \log \left (a+b (c+d x)^3\right )}{3 a^2 d}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 0.79 \begin {gather*} \frac {b \log \left (a+b (c+d x)^3\right )-\frac {a}{(c+d x)^3}-3 b \log (c+d x)}{3 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^4*(a + b*(c + d*x)^3)),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((c + d*x)^4*(a + b*(c + d*x)^3)),x]

[Out]

IntegrateAlgebraic[1/((c + d*x)^4*(a + b*(c + d*x)^3)), x]

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fricas [B]  time = 1.19, size = 158, normalized size = 2.82 \begin {gather*} \frac {{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (d x + c\right ) - a}{3 \, {\left (a^{2} d^{4} x^{3} + 3 \, a^{2} c d^{3} x^{2} + 3 \, a^{2} c^{2} d^{2} x + a^{2} c^{3} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3),x, algorithm="fricas")

[Out]

1/3*((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a
) - 3*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(d*x + c) - a)/(a^2*d^4*x^3 + 3*a^2*c*d^3*x^2 + 3*a
^2*c^2*d^2*x + a^2*c^3*d)

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giac [A]  time = 0.21, size = 41, normalized size = 0.73 \begin {gather*} \frac {b \log \left ({\left | -b - \frac {a}{{\left (d x + c\right )}^{3}} \right |}\right )}{3 \, a^{2} d} - \frac {1}{3 \, {\left (d x + c\right )}^{3} a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3),x, algorithm="giac")

[Out]

1/3*b*log(abs(-b - a/(d*x + c)^3))/(a^2*d) - 1/3/((d*x + c)^3*a*d)

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maple [A]  time = 0.01, size = 75, normalized size = 1.34 \begin {gather*} -\frac {b \ln \left (d x +c \right )}{a^{2} d}+\frac {b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a^{2} d}-\frac {1}{3 \left (d x +c \right )^{3} a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^4/(a+b*(d*x+c)^3),x)

[Out]

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

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maxima [A]  time = 0.59, size = 98, normalized size = 1.75 \begin {gather*} -\frac {1}{3 \, {\left (a d^{4} x^{3} + 3 \, a c d^{3} x^{2} + 3 \, a c^{2} d^{2} x + a c^{3} d\right )}} + \frac {b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d} - \frac {b \log \left (d x + c\right )}{a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3),x, algorithm="maxima")

[Out]

-1/3/(a*d^4*x^3 + 3*a*c*d^3*x^2 + 3*a*c^2*d^2*x + a*c^3*d) + 1/3*b*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x
+ b*c^3 + a)/(a^2*d) - b*log(d*x + c)/(a^2*d)

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mupad [B]  time = 1.42, size = 104, normalized size = 1.86 \begin {gather*} \frac {b\,\ln \left (-b\,c^3-3\,b\,c^2\,d\,x-3\,b\,c\,d^2\,x^2-b\,d^3\,x^3-a\right )}{3\,a^2\,d}-\frac {b\,\ln \left (c+d\,x\right )}{a^2\,d}-\frac {1}{3\,\left (a\,c^3\,d+3\,a\,c^2\,d^2\,x+3\,a\,c\,d^3\,x^2+a\,d^4\,x^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*(c + d*x)^3)*(c + d*x)^4),x)

[Out]

(b*log(- a - b*c^3 - b*d^3*x^3 - 3*b*c^2*d*x - 3*b*c*d^2*x^2))/(3*a^2*d) - (b*log(c + d*x))/(a^2*d) - 1/(3*(a*
d^4*x^3 + a*c^3*d + 3*a*c^2*d^2*x + 3*a*c*d^3*x^2))

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sympy [B]  time = 1.28, size = 100, normalized size = 1.79 \begin {gather*} - \frac {1}{3 a c^{3} d + 9 a c^{2} d^{2} x + 9 a c d^{3} x^{2} + 3 a d^{4} x^{3}} - \frac {b \log {\left (\frac {c}{d} + x \right )}}{a^{2} d} + \frac {b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**4/(a+b*(d*x+c)**3),x)

[Out]

-1/(3*a*c**3*d + 9*a*c**2*d**2*x + 9*a*c*d**3*x**2 + 3*a*d**4*x**3) - b*log(c/d + x)/(a**2*d) + b*log(3*c**2*x
/d**2 + 3*c*x**2/d + x**3 + (a + b*c**3)/(b*d**3))/(3*a**2*d)

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