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

Optimal. Leaf size=197 \[ -\frac{(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac{(5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{x (5 a d+b c)}{18 c^2 d \left (c+d x^3\right )}-\frac{x (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

[Out]

-((b*c - a*d)*x)/(6*c*d*(c + d*x^3)^2) + ((b*c + 5*a*d)*x)/(18*c^2*d*(c + d*x^3)) - ((b*c + 5*a*d)*ArcTan[(c^(
1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]*c^(8/3)*d^(4/3)) + ((b*c + 5*a*d)*Log[c^(1/3) + d^(1/3)*x])
/(27*c^(8/3)*d^(4/3)) - ((b*c + 5*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(4/3))

________________________________________________________________________________________

Rubi [A]  time = 0.106395, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {385, 199, 200, 31, 634, 617, 204, 628} \[ -\frac{(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac{(5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{x (5 a d+b c)}{18 c^2 d \left (c+d x^3\right )}-\frac{x (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)/(c + d*x^3)^3,x]

[Out]

-((b*c - a*d)*x)/(6*c*d*(c + d*x^3)^2) + ((b*c + 5*a*d)*x)/(18*c^2*d*(c + d*x^3)) - ((b*c + 5*a*d)*ArcTan[(c^(
1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]*c^(8/3)*d^(4/3)) + ((b*c + 5*a*d)*Log[c^(1/3) + d^(1/3)*x])
/(27*c^(8/3)*d^(4/3)) - ((b*c + 5*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(4/3))

Rule 385

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 31

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

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]

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 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 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]

Rubi steps

\begin{align*} \int \frac{a+b x^3}{\left (c+d x^3\right )^3} \, dx &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) \int \frac{1}{\left (c+d x^3\right )^2} \, dx}{6 c d}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac{(b c+5 a d) \int \frac{1}{c+d x^3} \, dx}{9 c^2 d}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac{(b c+5 a d) \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{27 c^{8/3} d}+\frac{(b c+5 a d) \int \frac{2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{27 c^{8/3} d}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac{(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(b c+5 a d) \int \frac{-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{54 c^{8/3} d^{4/3}}+\frac{(b c+5 a d) \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{18 c^{7/3} d}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac{(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(b c+5 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac{(b c+5 a d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{9 c^{8/3} d^{4/3}}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}-\frac{(b c+5 a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(b c+5 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}\\ \end{align*}

Mathematica [A]  time = 0.131517, size = 175, normalized size = 0.89 \[ \frac{-(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-\frac{9 c^{5/3} \sqrt [3]{d} x (b c-a d)}{\left (c+d x^3\right )^2}+\frac{3 c^{2/3} \sqrt [3]{d} x (5 a d+b c)}{c+d x^3}+2 (5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-2 \sqrt{3} (5 a d+b c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{54 c^{8/3} d^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-9*c^(5/3)*d^(1/3)*(b*c - a*d)*x)/(c + d*x^3)^2 + (3*c^(2/3)*d^(1/3)*(b*c + 5*a*d)*x)/(c + d*x^3) - 2*Sqrt[3
]*(b*c + 5*a*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(b*c + 5*a*d)*Log[c^(1/3) + d^(1/3)*x] - (b*c
+ 5*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(4/3))

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 249, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ({\frac{ \left ( 5\,ad+bc \right ){x}^{4}}{18\,{c}^{2}}}+{\frac{ \left ( 4\,ad-bc \right ) x}{9\,cd}} \right ) }+{\frac{5\,a}{27\,{c}^{2}d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{27\,c{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,a}{54\,{c}^{2}d}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{54\,c{d}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}a}{27\,{c}^{2}d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{27\,c{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/18*(5*a*d+b*c)/c^2*x^4+1/9*(4*a*d-b*c)/c/d*x)/(d*x^3+c)^2+5/27/c^2/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a+1/27/c
/d^2/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*b-5/54/c^2/d/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*a-1/54/c/d^2/(c/
d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*b+5/27/c^2/d/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*
x-1))*a+1/27/c/d^2/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*b

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.72292, size = 1648, normalized size = 8.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(3*(b*c^3*d^2 + 5*a*c^2*d^3)*x^4 + 3*sqrt(1/3)*((b*c^2*d^3 + 5*a*c*d^4)*x^6 + b*c^4*d + 5*a*c^3*d^2 + 2*
(b*c^3*d^2 + 5*a*c^2*d^3)*x^3)*sqrt(-(c^2*d)^(1/3)/d)*log((2*c*d*x^3 - 3*(c^2*d)^(1/3)*c*x - c^2 + 3*sqrt(1/3)
*(2*c*d*x^2 + (c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt(-(c^2*d)^(1/3)/d))/(d*x^3 + c)) - ((b*c*d^2 + 5*a*d^3)*x
^6 + b*c^3 + 5*a*c^2*d + 2*(b*c^2*d + 5*a*c*d^2)*x^3)*(c^2*d)^(2/3)*log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1
/3)*c) + 2*((b*c*d^2 + 5*a*d^3)*x^6 + b*c^3 + 5*a*c^2*d + 2*(b*c^2*d + 5*a*c*d^2)*x^3)*(c^2*d)^(2/3)*log(c*d*x
+ (c^2*d)^(2/3)) - 6*(b*c^4*d - 4*a*c^3*d^2)*x)/(c^4*d^4*x^6 + 2*c^5*d^3*x^3 + c^6*d^2), 1/54*(3*(b*c^3*d^2 +
5*a*c^2*d^3)*x^4 + 6*sqrt(1/3)*((b*c^2*d^3 + 5*a*c*d^4)*x^6 + b*c^4*d + 5*a*c^3*d^2 + 2*(b*c^3*d^2 + 5*a*c^2*
d^3)*x^3)*sqrt((c^2*d)^(1/3)/d)*arctan(sqrt(1/3)*(2*(c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt((c^2*d)^(1/3)/d)/c
^2) - ((b*c*d^2 + 5*a*d^3)*x^6 + b*c^3 + 5*a*c^2*d + 2*(b*c^2*d + 5*a*c*d^2)*x^3)*(c^2*d)^(2/3)*log(c*d*x^2 -
(c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) + 2*((b*c*d^2 + 5*a*d^3)*x^6 + b*c^3 + 5*a*c^2*d + 2*(b*c^2*d + 5*a*c*d^2)*
x^3)*(c^2*d)^(2/3)*log(c*d*x + (c^2*d)^(2/3)) - 6*(b*c^4*d - 4*a*c^3*d^2)*x)/(c^4*d^4*x^6 + 2*c^5*d^3*x^3 + c^
6*d^2)]

________________________________________________________________________________________

Sympy [A]  time = 1.36349, size = 133, normalized size = 0.68 \begin{align*} \frac{x^{4} \left (5 a d^{2} + b c d\right ) + x \left (8 a c d - 2 b c^{2}\right )}{18 c^{4} d + 36 c^{3} d^{2} x^{3} + 18 c^{2} d^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} c^{8} d^{4} - 125 a^{3} d^{3} - 75 a^{2} b c d^{2} - 15 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log{\left (\frac{27 t c^{3} d}{5 a d + b c} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x**4*(5*a*d**2 + b*c*d) + x*(8*a*c*d - 2*b*c**2))/(18*c**4*d + 36*c**3*d**2*x**3 + 18*c**2*d**3*x**6) + RootS
um(19683*_t**3*c**8*d**4 - 125*a**3*d**3 - 75*a**2*b*c*d**2 - 15*a*b**2*c**2*d - b**3*c**3, Lambda(_t, _t*log(
27*_t*c**3*d/(5*a*d + b*c) + x)))

________________________________________________________________________________________

Giac [A]  time = 1.14816, size = 273, normalized size = 1.39 \begin{align*} -\frac{{\left (b c + 5 \, a d\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{27 \, c^{3} d} + \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{27 \, c^{3} d^{2}} + \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{54 \, c^{3} d^{2}} + \frac{b c d x^{4} + 5 \, a d^{2} x^{4} - 2 \, b c^{2} x + 8 \, a c d x}{18 \,{\left (d x^{3} + c\right )}^{2} c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(b*c + 5*a*d)*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(c^3*d) + 1/27*sqrt(3)*((-c*d^2)^(1/3)*b*c + 5*(-c
*d^2)^(1/3)*a*d)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(c^3*d^2) + 1/54*((-c*d^2)^(1/3)*b*c +
5*(-c*d^2)^(1/3)*a*d)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c^3*d^2) + 1/18*(b*c*d*x^4 + 5*a*d^2*x^4 - 2*b
*c^2*x + 8*a*c*d*x)/((d*x^3 + c)^2*c^2*d)