3.15.53 \(\int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx\)

Optimal. Leaf size=219 \[ \frac {(b c-a d)^2 \log (c+d x)}{18 b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3} d^{5/3}}+\frac {(a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b} \]

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Rubi [A]  time = 0.09, antiderivative size = 219, 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 = {50, 59} \begin {gather*} \frac {(b c-a d)^2 \log (c+d x)}{18 b^{4/3} d^{5/3}}+\frac {(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 b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \tan ^{-1}\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} b^{4/3} d^{5/3}}+\frac {(a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)*(a + b*x)^(2/3)*(c + d*x)^(1/3))/(6*b*d) + ((a + b*x)^(5/3)*(c + d*x)^(1/3))/(2*b) + ((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^(4/3)*d^(5/
3)) + ((b*c - a*d)^2*Log[c + d*x])/(18*b^(4/3)*d^(5/3)) + ((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^(4/3)*d^(5/3))

Rule 50

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 59

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

Rubi steps

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

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Mathematica [C]  time = 0.03, size = 73, normalized size = 0.33 \begin {gather*} \frac {3 (a+b x)^{5/3} \sqrt [3]{c+d x} \, _2F_1\left (-\frac {1}{3},\frac {5}{3};\frac {8}{3};\frac {d (a+b x)}{a d-b c}\right )}{5 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(5/3)*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, 5/3, 8/3, (d*(a + b*x))/(-(b*c) + a*d)])/(5*b*((b*(
c + d*x))/(b*c - a*d))^(1/3))

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IntegrateAlgebraic [A]  time = 0.49, size = 294, normalized size = 1.34 \begin {gather*} \frac {(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 )}{9 b^{4/3} d^{5/3}}-\frac {(b c-a d)^2 \log \left (\frac {b^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+d^{2/3}\right )}{18 b^{4/3} d^{5/3}}-\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \left (\frac {b (c+d x)^{4/3}}{(a+b x)^{4/3}}+\frac {2 d \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )}{6 b d \left (\frac {b (c+d x)}{a+b x}-d\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.89, size = 717, normalized size = 3.27 \begin {gather*} \left [\frac {3 \, \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 ) + 2 \, {\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 ) - {\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 + b^{2} c d^{2} + 2 \, a b d^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{18 \, b^{2} d^{3}}, -\frac {6 \, \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 ) - 2 \, {\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 ) + {\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 + b^{2} c d^{2} + 2 \, a b d^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{18 \, b^{2} d^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/18*(3*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)) + 2*(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)) - (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 + b^2*c*d^2
+ 2*a*b*d^3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(b^2*d^3), -1/18*(6*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)) - 2*(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)) + (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 + b^2*c*d^2 + 2*a*b*d^3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(
b^2*d^3)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^(2/3)*(d*x + c)^(1/3), x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x +a \right )^{\frac {2}{3}} \left (d x +c \right )^{\frac {1}{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(2/3)*(d*x+c)^(1/3),x)

[Out]

int((b*x+a)^(2/3)*(d*x+c)^(1/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^(2/3)*(d*x + c)^(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(2/3)*(c + d*x)^(1/3),x)

[Out]

int((a + b*x)^(2/3)*(c + d*x)^(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*x)**(2/3)*(c + d*x)**(1/3), x)

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