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

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

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Rubi [A]  time = 0.07, 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 (a+b x)}{18 b^{5/3} d^{4/3}}+\frac {(b c-a d)^2 \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{4/3}}+\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^{5/3} d^{4/3}}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{3 b d}+\frac {(a+b x)^{4/3} (c+d x)^{2/3}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C]  time = 0.04, size = 73, normalized size = 0.33 \begin {gather*} \frac {3 (a+b x)^{4/3} (c+d x)^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {4}{3};\frac {7}{3};\frac {d (a+b x)}{a d-b c}\right )}{4 b \left (\frac {b (c+d x)}{b c-a d}\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(4/3)*(c + d*x)^(2/3)*Hypergeometric2F1[-2/3, 4/3, 7/3, (d*(a + b*x))/(-(b*c) + a*d)])/(4*b*((b*(
c + d*x))/(b*c - a*d))^(2/3))

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IntegrateAlgebraic [A]  time = 0.50, size = 293, normalized size = 1.34 \begin {gather*} \frac {(b c-a d)^2 \log \left (\sqrt [3]{b}-\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{9 b^{5/3} d^{4/3}}-\frac {(b c-a d)^2 \log \left (\frac {d^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+b^{2/3}\right )}{18 b^{5/3} d^{4/3}}-\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^{5/3} d^{4/3}}+\frac {(b c-a d)^2 \left (\frac {d (a+b x)^{4/3}}{(c+d x)^{4/3}}+\frac {2 b \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{6 b d \left (b-\frac {d (a+b x)}{c+d x}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.22, size = 716, 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^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (3 \, b^{2} d x + b^{2} c + 2 \, a b d - 3 \, \left (b^{2} d\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {-\frac {\left (b^{2} d\right )^{\frac {1}{3}}}{d}}\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (b^{2} d\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^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \left (b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (b^{2} d\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^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right ) + 3 \, {\left (3 \, b^{3} d^{2} x + 2 \, b^{3} c d + a b^{2} d^{2}\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, b^{3} d^{2}}, -\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^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \left (b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {\frac {\left (b^{2} d\right )^{\frac {1}{3}}}{d}}}{b^{2} d x + b^{2} c}\right ) + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (b^{2} d\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^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \left (b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (b^{2} d\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^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right ) - 3 \, {\left (3 \, b^{3} d^{2} x + 2 \, b^{3} c d + a b^{2} d^{2}\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, b^{3} d^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/3)*(d*x+c)^(2/3),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^2*d)^(1/3)/d)*log(3*b^2*d*x + b^2*c + 2*a*
b*d - 3*(b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b - 3*sqrt(1/3)*(2*(b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d -
(b^2*d)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (b^2*d)^(1/3)*(b*d*x + b*c))*sqrt(-(b^2*d)^(1/3)/d)) - (b^2*c
^2 - 2*a*b*c*d + a^2*d^2)*(b^2*d)^(2/3)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d + (b^2*d)^(2/3)*(b*x + a)^(1/
3)*(d*x + c)^(2/3) + (b^2*d)^(1/3)*(b*d*x + b*c))/(d*x + c)) + 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b^2*d)^(2/3)
*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d - (b^2*d)^(2/3)*(d*x + c))/(d*x + c)) + 3*(3*b^3*d^2*x + 2*b^3*c*d +
a*b^2*d^2)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^3*d^2), -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^2*d)^(1/3)/d)*arctan(sqrt(1/3)*(2*(b^2*d)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (b^2*d)^(1/3)*(
b*d*x + b*c))*sqrt((b^2*d)^(1/3)/d)/(b^2*d*x + b^2*c)) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b^2*d)^(2/3)*log(((b
*x + a)^(2/3)*(d*x + c)^(1/3)*b*d + (b^2*d)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (b^2*d)^(1/3)*(b*d*x + b*c
))/(d*x + c)) - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b^2*d)^(2/3)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d - (b^
2*d)^(2/3)*(d*x + c))/(d*x + c)) - 3*(3*b^3*d^2*x + 2*b^3*c*d + a*b^2*d^2)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b
^3*d^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*x+a)^(1/3)*(d*x+c)^(2/3),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 {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{a + b x} \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)**(1/3)*(d*x+c)**(2/3),x)

[Out]

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

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