3.31.17 \(\int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx\) [3017]

Optimal. Leaf size=200 \[ \frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d}-\frac {(3 b d e-2 b c f-a d f) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt {3} b^{4/3} d^{5/3}}-\frac {(3 b d e-2 b c f-a d f) \log (c+d x)}{6 b^{4/3} d^{5/3}}-\frac {(3 b d e-2 b c f-a d f) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3} d^{5/3}} \]

[Out]

f*(b*x+a)^(2/3)*(d*x+c)^(1/3)/b/d-1/6*(-a*d*f-2*b*c*f+3*b*d*e)*ln(d*x+c)/b^(4/3)/d^(5/3)-1/2*(-a*d*f-2*b*c*f+3
*b*d*e)*ln(-1+d^(1/3)*(b*x+a)^(1/3)/b^(1/3)/(d*x+c)^(1/3))/b^(4/3)/d^(5/3)-1/3*(-a*d*f-2*b*c*f+3*b*d*e)*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^(4/3)/d^(5/3)*3^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {81, 61} \begin {gather*} -\frac {\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 ) (-a d f-2 b c f+3 b d e)}{\sqrt {3} b^{4/3} d^{5/3}}-\frac {\log (c+d x) (-a d f-2 b c f+3 b d e)}{6 b^{4/3} d^{5/3}}-\frac {(-a d f-2 b c f+3 b d e) \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{5/3}}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rubi steps

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

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Mathematica [A]
time = 0.44, size = 234, normalized size = 1.17 \begin {gather*} \frac {6 \sqrt [3]{b} d^{2/3} f (a+b x)^{2/3} \sqrt [3]{c+d x}+2 \sqrt {3} (3 b d e-2 b c f-a d f) \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 )+2 (-3 b d e+2 b c f+a d f) \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )+(3 b d e-2 b c f-a d f) \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 )}{6 b^{4/3} d^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {f x +e}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x+e)/(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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.91, size = 687, normalized size = 3.44 \begin {gather*} \left [\frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} f + 3 \, \sqrt {\frac {1}{3}} {\left (3 \, b^{2} d^{2} e - {\left (2 \, b^{2} c d + a b d^{2}\right )} f\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 d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \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 d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \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 )}{6 \, b^{2} d^{3}}, \frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} f + 6 \, \sqrt {\frac {1}{3}} {\left (3 \, b^{2} d^{2} e - {\left (2 \, b^{2} c d + a b d^{2}\right )} f\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 d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \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 d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \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 )}{6 \, b^{2} d^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

integrate((f*x + e)/((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 \frac {e+f\,x}{{\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((e + f*x)/((a + b*x)^(1/3)*(c + d*x)^(2/3)),x)

[Out]

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

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