3.7.96 \(\int \frac {1}{x^{4/3} (a+b x)^3} \, dx\)

Optimal. Leaf size=152 \[ \frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {51, 56, 617, 204, 31} \begin {gather*} \frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(4/3)*(a + b*x)^3),x]

[Out]

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

Rule 31

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

Rule 51

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 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[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] && Ne
gQ[(b*c - a*d)/b]

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 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]

Rubi steps

\begin {align*} \int \frac {1}{x^{4/3} (a+b x)^3} \, dx &=\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7 \int \frac {1}{x^{4/3} (a+b x)^2} \, dx}{6 a}\\ &=\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {14 \int \frac {1}{x^{4/3} (a+b x)} \, dx}{9 a^2}\\ &=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}-\frac {(14 b) \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{9 a^3}\\ &=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}-\frac {7 \operatorname {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 )}{3 a^3}+\frac {\left (7 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{3 a^{10/3}}\\ &=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}-\frac {\left (14 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{3 a^{10/3}}\\ &=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 25, normalized size = 0.16 \begin {gather*} -\frac {3 \, _2F_1\left (-\frac {1}{3},3;\frac {2}{3};-\frac {b x}{a}\right )}{a^3 \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(4/3)*(a + b*x)^3),x]

[Out]

(-3*Hypergeometric2F1[-1/3, 3, 2/3, -((b*x)/a)])/(a^3*x^(1/3))

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IntegrateAlgebraic [A]  time = 0.27, size = 168, normalized size = 1.11 \begin {gather*} -\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{9 a^{10/3}}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{9 a^{10/3}}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {-18 a^2-49 a b x-28 b^2 x^2}{6 a^3 \sqrt [3]{x} (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*a^2 - 49*a*b*x - 28*b^2*x^2)/(6*a^3*x^(1/3)*(a + b*x)^2) + (14*b^(1/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*x^(1
/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(10/3)) + (14*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(9*a^(10/3)) - (7*
b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)])/(9*a^(10/3))

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fricas [A]  time = 1.29, size = 211, normalized size = 1.39 \begin {gather*} -\frac {28 \, \sqrt {3} {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} 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 ) + 14 \, {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} 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 ) - 28 \, {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} 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 (28 \, b^{2} x^{2} + 49 \, a b x + 18 \, a^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(28*sqrt(3)*(b^2*x^3 + 2*a*b*x^2 + a^2*x)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x^(1/3)*(b/a)^(1/3) - 1/3*sqrt(
3)) + 14*(b^2*x^3 + 2*a*b*x^2 + a^2*x)*(b/a)^(1/3)*log(-a*x^(1/3)*(b/a)^(2/3) + b*x^(2/3) + a*(b/a)^(1/3)) - 2
8*(b^2*x^3 + 2*a*b*x^2 + a^2*x)*(b/a)^(1/3)*log(a*(b/a)^(2/3) + b*x^(1/3)) + 3*(28*b^2*x^2 + 49*a*b*x + 18*a^2
)*x^(2/3))/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x)

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giac [A]  time = 1.16, size = 155, normalized size = 1.02 \begin {gather*} \frac {14 \, 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 )}{9 \, a^{4}} + \frac {14 \, \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 )}{9 \, a^{4} b} - \frac {3}{a^{3} x^{\frac {1}{3}}} - \frac {7 \, \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 )}{9 \, a^{4} b} - \frac {10 \, b^{2} x^{\frac {5}{3}} + 13 \, a b x^{\frac {2}{3}}}{6 \, {\left (b x + a\right )}^{2} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14/9*b*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^4 + 14/9*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^4*b) - 3/(a^3*x^(1/3)) - 7/9*(-a*b^2)^(2/3)*log(x^(2/3) + x^(1/3)*(-a/
b)^(1/3) + (-a/b)^(2/3))/(a^4*b) - 1/6*(10*b^2*x^(5/3) + 13*a*b*x^(2/3))/((b*x + a)^2*a^3)

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maple [A]  time = 0.02, size = 139, normalized size = 0.91 \begin {gather*} -\frac {5 b^{2} x^{\frac {5}{3}}}{3 \left (b x +a \right )^{2} a^{3}}-\frac {13 b \,x^{\frac {2}{3}}}{6 \left (b x +a \right )^{2} a^{2}}-\frac {14 \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 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {14 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 \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 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {3}{a^{3} x^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/a^3/x^(1/3)-5/3*b^2/a^3/(b*x+a)^2*x^(5/3)-13/6*b/a^2/(b*x+a)^2*x^(2/3)+14/9/a^3/(a/b)^(1/3)*ln(x^(1/3)+(a/b
)^(1/3))-7/9/a^3/(a/b)^(1/3)*ln(x^(2/3)-(a/b)^(1/3)*x^(1/3)+(a/b)^(2/3))-14/9/a^3*3^(1/2)/(a/b)^(1/3)*arctan(1
/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/3)-1))

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maxima [A]  time = 2.99, size = 154, normalized size = 1.01 \begin {gather*} -\frac {28 \, b^{2} x^{2} + 49 \, a b x + 18 \, a^{2}}{6 \, {\left (a^{3} b^{2} x^{\frac {7}{3}} + 2 \, a^{4} b x^{\frac {4}{3}} + a^{5} x^{\frac {1}{3}}\right )}} - \frac {14 \, \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 )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {7 \, \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 )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {14 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(28*b^2*x^2 + 49*a*b*x + 18*a^2)/(a^3*b^2*x^(7/3) + 2*a^4*b*x^(4/3) + a^5*x^(1/3)) - 14/9*sqrt(3)*arctan(
1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*(a/b)^(1/3)) - 7/9*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) +
(a/b)^(2/3))/(a^3*(a/b)^(1/3)) + 14/9*log(x^(1/3) + (a/b)^(1/3))/(a^3*(a/b)^(1/3))

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mupad [B]  time = 0.09, size = 174, normalized size = 1.14 \begin {gather*} \frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}+588\,a^3\,b^3\,x^{1/3}\right )}{9\,a^{10/3}}-\frac {\frac {3}{a}+\frac {14\,b^2\,x^2}{3\,a^3}+\frac {49\,b\,x}{6\,a^2}}{a^2\,x^{1/3}+b^2\,x^{7/3}+2\,a\,b\,x^{4/3}}+\frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+588\,a^3\,b^3\,x^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{10/3}}-\frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+588\,a^3\,b^3\,x^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{10/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(4/3)*(a + b*x)^3),x)

[Out]

(14*b^(1/3)*log(588*a^(10/3)*b^(8/3) + 588*a^3*b^3*x^(1/3)))/(9*a^(10/3)) - (3/a + (14*b^2*x^2)/(3*a^3) + (49*
b*x)/(6*a^2))/(a^2*x^(1/3) + b^2*x^(7/3) + 2*a*b*x^(4/3)) + (14*b^(1/3)*log(588*a^(10/3)*b^(8/3)*((3^(1/2)*1i)
/2 - 1/2)^2 + 588*a^3*b^3*x^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(9*a^(10/3)) - (14*b^(1/3)*log(588*a^(10/3)*b^(8/3)
*((3^(1/2)*1i)/2 + 1/2)^2 + 588*a^3*b^3*x^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(9*a^(10/3))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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