\(\int \frac {1}{x^{4/3} (a+b x)^2} \, dx\) [688]

Optimal result

Integrand size = 13, antiderivative size = 124 \[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx=-\frac {4}{a^2 \sqrt [3]{x}}+\frac {1}{a \sqrt [3]{x} (a+b x)}+\frac {4 \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{7/3}}-\frac {2 \sqrt [3]{b} \log (a+b x)}{3 a^{7/3}} \]

[Out]

-4/a^2/x^(1/3)+1/a/x^(1/3)/(b*x+a)+2*b^(1/3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/a^(7/3)-2/3*b^(1/3)*ln(b*x+a)/a^(7/3)
+4/3*b^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3))/a^(1/3)*3^(1/2))/a^(7/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {44, 53, 58, 631, 210, 31} \[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx=\frac {4 \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{7/3}}-\frac {2 \sqrt [3]{b} \log (a+b x)}{3 a^{7/3}}-\frac {4}{a^2 \sqrt [3]{x}}+\frac {1}{a \sqrt [3]{x} (a+b x)} \]

[In]

Int[1/(x^(4/3)*(a + b*x)^2),x]

[Out]

-4/(a^2*x^(1/3)) + 1/(a*x^(1/3)*(a + b*x)) + (4*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))
])/(Sqrt[3]*a^(7/3)) + (2*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/a^(7/3) - (2*b^(1/3)*Log[a + b*x])/(3*a^(7/3
))

Rule 31

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

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ
[(b*c - a*d)/b]

Rule 210

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

Rule 631

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{a \sqrt [3]{x} (a+b x)}+\frac {4 \int \frac {1}{x^{4/3} (a+b x)} \, dx}{3 a} \\ & = -\frac {4}{a^2 \sqrt [3]{x}}+\frac {1}{a \sqrt [3]{x} (a+b x)}-\frac {(4 b) \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{3 a^2} \\ & = -\frac {4}{a^2 \sqrt [3]{x}}+\frac {1}{a \sqrt [3]{x} (a+b x)}-\frac {2 \sqrt [3]{b} \log (a+b x)}{3 a^{7/3}}-\frac {2 \text {Subst}\left (\int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{a^2}+\frac {\left (2 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{a^{7/3}} \\ & = -\frac {4}{a^2 \sqrt [3]{x}}+\frac {1}{a \sqrt [3]{x} (a+b x)}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{7/3}}-\frac {2 \sqrt [3]{b} \log (a+b x)}{3 a^{7/3}}-\frac {\left (4 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{7/3}} \\ & = -\frac {4}{a^2 \sqrt [3]{x}}+\frac {1}{a \sqrt [3]{x} (a+b x)}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{7/3}}-\frac {2 \sqrt [3]{b} \log (a+b x)}{3 a^{7/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx=\frac {-\frac {3 \sqrt [3]{a} (3 a+4 b x)}{\sqrt [3]{x} (a+b x)}+4 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{3 a^{7/3}} \]

[In]

Integrate[1/(x^(4/3)*(a + b*x)^2),x]

[Out]

((-3*a^(1/3)*(3*a + 4*b*x))/(x^(1/3)*(a + b*x)) + 4*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/S
qrt[3]] + 4*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)] - 2*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)
*x^(2/3)])/(3*a^(7/3))

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {3}{a^{2} x^{\frac {1}{3}}}-\frac {b \,x^{\frac {2}{3}}}{a^{2} \left (b x +a \right )}+\frac {4 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {2 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}\)	\(121\)
derivativedivides	\(-\frac {3}{a^{2} x^{\frac {1}{3}}}-\frac {3 b \left (\frac {x^{\frac {2}{3}}}{3 b x +3 a}-\frac {4 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2}}\)	\(124\)
default	\(-\frac {3}{a^{2} x^{\frac {1}{3}}}-\frac {3 b \left (\frac {x^{\frac {2}{3}}}{3 b x +3 a}-\frac {4 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2}}\)	\(124\)

[In]

int(1/x^(4/3)/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-3/a^2/x^(1/3)-b/a^2*x^(2/3)/(b*x+a)+4/3/a^2/(a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/3))-2/3/a^2/(a/b)^(1/3)*ln(x^(2/3
)-(a/b)^(1/3)*x^(1/3)+(a/b)^(2/3))-4/3/a^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/3)-1))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx=-\frac {4 \, \sqrt {3} {\left (b x^{2} + a x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, {\left (b x^{2} + a x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (-a x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 4 \, {\left (b x^{2} + a x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (a \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, b x + 3 \, a\right )} x^{\frac {2}{3}}}{3 \, {\left (a^{2} b x^{2} + a^{3} x\right )}} \]

[In]

integrate(1/x^(4/3)/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/3*(4*sqrt(3)*(b*x^2 + a*x)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x^(1/3)*(b/a)^(1/3) - 1/3*sqrt(3)) + 2*(b*x^2 + a
*x)*(b/a)^(1/3)*log(-a*x^(1/3)*(b/a)^(2/3) + b*x^(2/3) + a*(b/a)^(1/3)) - 4*(b*x^2 + a*x)*(b/a)^(1/3)*log(a*(b
/a)^(2/3) + b*x^(1/3)) + 3*(4*b*x + 3*a)*x^(2/3))/(a^2*b*x^2 + a^3*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx=\text {Timed out} \]

[In]

integrate(1/x**(4/3)/(b*x+a)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx=-\frac {4 \, b x + 3 \, a}{a^{2} b x^{\frac {4}{3}} + a^{3} x^{\frac {1}{3}}} - \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {4 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate(1/x^(4/3)/(b*x+a)^2,x, algorithm="maxima")

[Out]

-(4*b*x + 3*a)/(a^2*b*x^(4/3) + a^3*x^(1/3)) - 4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^
(1/3))/(a^2*(a/b)^(1/3)) - 2/3*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*(a/b)^(1/3)) + 4/3*log(x^
(1/3) + (a/b)^(1/3))/(a^2*(a/b)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx=\frac {4 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{3}} + \frac {4 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3} b} - \frac {4 \, b x + 3 \, a}{{\left (b x^{\frac {4}{3}} + a x^{\frac {1}{3}}\right )} a^{2}} - \frac {2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, a^{3} b} \]

[In]

integrate(1/x^(4/3)/(b*x+a)^2,x, algorithm="giac")

[Out]

4/3*b*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^3 + 4/3*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x^(
1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b) - (4*b*x + 3*a)/((b*x^(4/3) + a*x^(1/3))*a^2) - 2/3*(-a*b^2)^(2/3)*
log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^{4/3} (a+b x)^2} \, dx=\frac {4\,b^{1/3}\,\ln \left (16\,a^{7/3}\,b^{8/3}+16\,a^2\,b^3\,x^{1/3}\right )}{3\,a^{7/3}}-\frac {\frac {3}{a}+\frac {4\,b\,x}{a^2}}{a\,x^{1/3}+b\,x^{4/3}}-\frac {4\,b^{1/3}\,\ln \left (16\,a^{7/3}\,b^{8/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+16\,a^2\,b^3\,x^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{7/3}}+\frac {b^{1/3}\,\ln \left (9\,a^{7/3}\,b^{8/3}\,{\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,a^2\,b^3\,x^{1/3}\right )\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}{a^{7/3}} \]

[In]

int(1/(x^(4/3)*(a + b*x)^2),x)

[Out]

(4*b^(1/3)*log(16*a^(7/3)*b^(8/3) + 16*a^2*b^3*x^(1/3)))/(3*a^(7/3)) - (3/a + (4*b*x)/a^2)/(a*x^(1/3) + b*x^(4
/3)) - (4*b^(1/3)*log(16*a^(7/3)*b^(8/3)*((3^(1/2)*1i)/2 + 1/2)^2 + 16*a^2*b^3*x^(1/3))*((3^(1/2)*1i)/2 + 1/2)
)/(3*a^(7/3)) + (b^(1/3)*log(9*a^(7/3)*b^(8/3)*((3^(1/2)*2i)/3 - 2/3)^2 + 16*a^2*b^3*x^(1/3))*((3^(1/2)*2i)/3
- 2/3))/a^(7/3)