3.17.1 \(\int \frac {(a+b x)^{5/3}}{(c+d x)^{2/3}} \, dx\) [1601]

Optimal. Leaf size=216 \[ -\frac {5 (b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 d^2}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d}-\frac {5 (b c-a d)^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{3 \sqrt {3} \sqrt [3]{b} d^{8/3}}-\frac {5 (b c-a d)^2 \log (c+d x)}{18 \sqrt [3]{b} d^{8/3}}-\frac {5 (b c-a d)^2 \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{6 \sqrt [3]{b} d^{8/3}} \]

[Out]

-5/6*(-a*d+b*c)*(b*x+a)^(2/3)*(d*x+c)^(1/3)/d^2+1/2*(b*x+a)^(5/3)*(d*x+c)^(1/3)/d-5/18*(-a*d+b*c)^2*ln(d*x+c)/
b^(1/3)/d^(8/3)-5/6*(-a*d+b*c)^2*ln(-1+d^(1/3)*(b*x+a)^(1/3)/b^(1/3)/(d*x+c)^(1/3))/b^(1/3)/d^(8/3)-5/9*(-a*d+
b*c)^2*arctan(1/3*3^(1/2)+2/3*d^(1/3)*(b*x+a)^(1/3)/b^(1/3)/(d*x+c)^(1/3)*3^(1/2))/b^(1/3)/d^(8/3)*3^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {52, 61} \begin {gather*} -\frac {5 (b c-a d)^2 \text {ArcTan}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b} d^{8/3}}-\frac {5 (b c-a d)^2 \log (c+d x)}{18 \sqrt [3]{b} d^{8/3}}-\frac {5 (b c-a d)^2 \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 \sqrt [3]{b} d^{8/3}}-\frac {5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 d^2}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/3)/(c + d*x)^(2/3),x]

[Out]

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

Rule 52

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

Rule 61

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt
[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*
((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] /; FreeQ[{a, b, c, d}, x] && Ne
Q[b*c - a*d, 0] && PosQ[d/b]

Rubi steps

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

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Mathematica [A]
time = 0.41, size = 231, normalized size = 1.07 \begin {gather*} \frac {3 \sqrt [3]{b} d^{2/3} (a+b x)^{2/3} \sqrt [3]{c+d x} (-5 b c+8 a d+3 b d x)+10 \sqrt {3} (b c-a d)^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )-10 (b c-a d)^2 \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )+5 (b c-a d)^2 \log \left (d^{2/3}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {b^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{18 \sqrt [3]{b} d^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/3)/(c + d*x)^(2/3),x]

[Out]

(3*b^(1/3)*d^(2/3)*(a + b*x)^(2/3)*(c + d*x)^(1/3)*(-5*b*c + 8*a*d + 3*b*d*x) + 10*Sqrt[3]*(b*c - a*d)^2*ArcTa
n[(1 + (2*b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3)))/Sqrt[3]] - 10*(b*c - a*d)^2*Log[d^(1/3) - (b^(1/
3)*(c + d*x)^(1/3))/(a + b*x)^(1/3)] + 5*(b*c - a*d)^2*Log[d^(2/3) + (b^(1/3)*d^(1/3)*(c + d*x)^(1/3))/(a + b*
x)^(1/3) + (b^(2/3)*(c + d*x)^(2/3))/(a + b*x)^(2/3)])/(18*b^(1/3)*d^(8/3))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {5}{3}}}{\left (d x +c \right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/3)/(d*x+c)^(2/3),x)

[Out]

int((b*x+a)^(5/3)/(d*x+c)^(2/3),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/3)/(d*x+c)^(2/3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/3)/(d*x + c)^(2/3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (166) = 332\).
time = 1.00, size = 741, normalized size = 3.43 \begin {gather*} \left [\frac {15 \, \sqrt {\frac {1}{3}} {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} \sqrt {\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}} \log \left (-3 \, b d^{2} x - 2 \, b c d - a d^{2} - 3 \, \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} d - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}}\right ) - 10 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + 5 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right ) + 3 \, {\left (3 \, b^{2} d^{3} x - 5 \, b^{2} c d^{2} + 8 \, a b d^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{18 \, b d^{4}}, \frac {30 \, \sqrt {\frac {1}{3}} {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} \sqrt {-\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {-\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}}}{b d^{2} x + a d^{2}}\right ) - 10 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + 5 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right ) + 3 \, {\left (3 \, b^{2} d^{3} x - 5 \, b^{2} c d^{2} + 8 \, a b d^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{18 \, b d^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/3)/(d*x+c)^(2/3),x, algorithm="fricas")

[Out]

[1/18*(15*sqrt(1/3)*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*sqrt((-b*d^2)^(1/3)/b)*log(-3*b*d^2*x - 2*b*c*d -
a*d^2 - 3*(-b*d^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3)*d - 3*sqrt(1/3)*(2*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b*
d - (-b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (-b*d^2)^(1/3)*(b*d*x + a*d))*sqrt((-b*d^2)^(1/3)/b)) - 1
0*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(-b*d^2)^(2/3)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d - (-b*d^2)^(2/3)*(b*
x + a))/(b*x + a)) + 5*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(-b*d^2)^(2/3)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d
+ (-b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2)^(1/3)*(b*d*x + a*d))/(b*x + a)) + 3*(3*b^2*d^3*x
- 5*b^2*c*d^2 + 8*a*b*d^3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*d^4), 1/18*(30*sqrt(1/3)*(b^3*c^2*d - 2*a*b^2*c
*d^2 + a^2*b*d^3)*sqrt(-(-b*d^2)^(1/3)/b)*arctan(sqrt(1/3)*(2*(-b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) -
(-b*d^2)^(1/3)*(b*d*x + a*d))*sqrt(-(-b*d^2)^(1/3)/b)/(b*d^2*x + a*d^2)) - 10*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)
*(-b*d^2)^(2/3)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d - (-b*d^2)^(2/3)*(b*x + a))/(b*x + a)) + 5*(b^2*c^2 -
2*a*b*c*d + a^2*d^2)*(-b*d^2)^(2/3)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d + (-b*d^2)^(2/3)*(b*x + a)^(2/3)
*(d*x + c)^(1/3) - (-b*d^2)^(1/3)*(b*d*x + a*d))/(b*x + a)) + 3*(3*b^2*d^3*x - 5*b^2*c*d^2 + 8*a*b*d^3)*(b*x +
a)^(2/3)*(d*x + c)^(1/3))/(b*d^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {5}{3}}}{\left (c + d x\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/3)/(d*x+c)**(2/3),x)

[Out]

Integral((a + b*x)**(5/3)/(c + d*x)**(2/3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/3)/(d*x+c)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/3)/(d*x + c)^(2/3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/3}}{{\left (c+d\,x\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(5/3)/(c + d*x)^(2/3),x)

[Out]

int((a + b*x)^(5/3)/(c + d*x)^(2/3), x)

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