3.3017 \(\int \frac{e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx\)

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

[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))

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Rubi [A]  time = 0.107498, antiderivative size = 200, normalized size of antiderivative = 1., 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 = {80, 59} \[ -\frac{\log (c+d x) (-a d f-2 b c f+3 b d e)}{6 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{5/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d} \]

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 80

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]

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 \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*}

Mathematica [C]  time = 0.110734, size = 103, normalized size = 0.52 \[ \frac{(a+b x)^{2/3} \left (\left (\frac{b (c+d x)}{b c-a d}\right )^{2/3} (-a d f-2 b c f+3 b d e) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{d (a+b x)}{a d-b c}\right )+2 b f (c+d x)\right )}{2 b^2 d (c+d x)^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(2/3)*(2*b*f*(c + d*x) + (3*b*d*e - 2*b*c*f - a*d*f)*((b*(c + d*x))/(b*c - a*d))^(2/3)*Hypergeometr
ic2F1[2/3, 2/3, 5/3, (d*(a + b*x))/(-(b*c) + a*d)]))/(2*b^2*d*(c + d*x)^(2/3))

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{(fx+e){\frac{1}{\sqrt [3]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

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 = 2.10227, size = 1693, normalized size = 8.46 \begin{align*} \text{result too large to display} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{e + f x}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

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)