3.3005 \(\int \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx\)

Optimal. Leaf size=417 \[ -\frac{3 \sqrt [3]{b} d^{2/3} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 f^2}-\frac{\sqrt{3} \sqrt [3]{b} d^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{f^2}-\frac{\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac{(-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac{(-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f (e+f x)}-\frac{\sqrt [3]{b} d^{2/3} \log (a+b x)}{2 f^2} \]

[Out]

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

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Rubi [A]  time = 0.266042, antiderivative size = 417, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {97, 157, 59, 91} \[ -\frac{3 \sqrt [3]{b} d^{2/3} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 f^2}-\frac{\sqrt{3} \sqrt [3]{b} d^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{f^2}-\frac{\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac{(-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac{(-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f (e+f x)}-\frac{\sqrt [3]{b} d^{2/3} \log (a+b x)}{2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 97

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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]

Rule 91

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

Rubi steps

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

Mathematica [A]  time = 2.3203, size = 504, normalized size = 1.21 \[ \frac{(c+d x)^{2/3} \left (\left (\frac{d (a+b x)}{c+d x}\right )^{2/3} \left (3 \sqrt [3]{b} \left (\log \left (\sqrt [3]{b} \sqrt [3]{\frac{d (a+b x)}{c+d x}}+\left (\frac{d (a+b x)}{c+d x}\right )^{2/3}+b^{2/3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{\frac{d (a+b x)}{c+d x}}}{\sqrt [3]{b}}+1}{\sqrt{3}}\right )\right )+\frac{2 (-2 a d f-b c f+3 b d e) \log \left (\sqrt [3]{d} \sqrt [3]{b e-a f}-\sqrt [3]{d e-c f} \sqrt [3]{\frac{d (a+b x)}{c+d x}}\right )}{d^{2/3} (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac{(-2 a d f-b c f+3 b d e) \left (\log \left (\sqrt [3]{d} \sqrt [3]{b e-a f} \sqrt [3]{d e-c f} \sqrt [3]{\frac{d (a+b x)}{c+d x}}+(d e-c f)^{2/3} \left (\frac{d (a+b x)}{c+d x}\right )^{2/3}+d^{2/3} (b e-a f)^{2/3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{d e-c f} \sqrt [3]{\frac{d (a+b x)}{c+d x}}}{\sqrt [3]{d} \sqrt [3]{b e-a f}}+1}{\sqrt{3}}\right )\right )}{d^{2/3} (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-6 \sqrt [3]{b} \log \left (\sqrt [3]{b}-\sqrt [3]{\frac{d (a+b x)}{c+d x}}\right )\right )-\frac{6 f (a+b x)}{e+f x}\right )}{6 f^2 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{2}}\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((b*x+a)^(1/3)*(d*x+c)^(2/3)/(f*x+e)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 42.976, size = 7911, normalized size = 18.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*sqrt(1/3)*(3*b^2*d^2*e^4 - (4*b^2*c*d + 5*a*b*d^2)*e^3*f + (b^2*c^2 + 6*a*b*c*d + 2*a^2*d^2)*e^2*f^2 -
(a*b*c^2 + 2*a^2*c*d)*e*f^3 + (3*b^2*d^2*e^3*f - (4*b^2*c*d + 5*a*b*d^2)*e^2*f^2 + (b^2*c^2 + 6*a*b*c*d + 2*a
^2*d^2)*e*f^3 - (a*b*c^2 + 2*a^2*c*d)*f^4)*x)*sqrt((-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*
c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f))*log(-(3*a^2*c*f^2 + (b^2*c + 2*a*b*d)*e^2 - 2*(2*a*b*c + a^2*d)*e*f + 3*(
-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*e - a*f)*(b*x + a)^(1/3)*
(d*x + c)^(2/3) + (3*b^2*d*e^2 - 2*(b^2*c + 2*a*b*d)*e*f + (2*a*b*c + a^2*d)*f^2)*x - 3*sqrt(1/3)*(2*(b*d*e^2
+ a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2
*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b
*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))*sqrt((-b^2*d*e^3 + a^2*c*f^3 +
(b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f)))/(f*x + e)) - 6*sqrt(3)*(b^2*d*e^4 - a^
2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2 + (b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)
*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*arctan(1/3*(2*sqrt(3)*(-b*d^2)^(2/3)*(b*x + a)^(1/3)*(d*
x + c)^(2/3) + sqrt(3)*(b*d^2*x + b*c*d))/(b*d^2*x + b*c*d)) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2
*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(3*b*d*e^2 - (b*c + 2*a*d)*e*f + (3*b*d*e*f - (b*c + 2*a*d)*f^2)*x)*log(((
b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*
b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c
+ 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))/(d*x + c)) + 2*(-b^2*d
*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(3*b*d*e^2 - (b*c + 2*a*d)*e*f + (
3*b*d*e*f - (b*c + 2*a*d)*f^2)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) -
(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(d*x + c))/(d*x + c)) - 3*
(b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2 + (b^2*d*e^3*f - a^2*c*f^4 - (b
^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*d^2
- (-b*d^2)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*d + (-b*d^2)^(2/3)*(d*x + c))/(d*x + c)) + 6*(b^2*d*e^4 - a^2
*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2 + (b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*
e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*d + (-b*d^2)^(1/3)*(
d*x + c))/(d*x + c)) - 6*(b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*(b*x
+ a)^(1/3)*(d*x + c)^(2/3))/(b^2*d*e^4*f^2 - a^2*c*e*f^5 - (b^2*c + 2*a*b*d)*e^3*f^3 + (2*a*b*c + a^2*d)*e^2*f
^4 + (b^2*d*e^3*f^3 - a^2*c*f^6 - (b^2*c + 2*a*b*d)*e^2*f^4 + (2*a*b*c + a^2*d)*e*f^5)*x), -1/6*(6*sqrt(1/3)*(
3*b^2*d^2*e^4 - (4*b^2*c*d + 5*a*b*d^2)*e^3*f + (b^2*c^2 + 6*a*b*c*d + 2*a^2*d^2)*e^2*f^2 - (a*b*c^2 + 2*a^2*c
*d)*e*f^3 + (3*b^2*d^2*e^3*f - (4*b^2*c*d + 5*a*b*d^2)*e^2*f^2 + (b^2*c^2 + 6*a*b*c*d + 2*a^2*d^2)*e*f^3 - (a*
b*c^2 + 2*a^2*c*d)*f^4)*x)*sqrt(-(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^
(1/3)/(d*e - c*f))*arctan(sqrt(1/3)*(2*(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e
*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a
^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))*sqrt(-(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*
f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f))/(b^2*c*e^2 - 2*a*b*c*e*f + a^2*c*f^2 + (b^2*d*e^2 - 2*a*b*d*e*
f + a^2*d*f^2)*x)) + 6*sqrt(3)*(b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2
+ (b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*arctan(1/3
*(2*sqrt(3)*(-b*d^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + sqrt(3)*(b*d^2*x + b*c*d))/(b*d^2*x + b*c*d)) + (
-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(3*b*d*e^2 - (b*c + 2*a*d)*e
*f + (3*b*d*e*f - (b*c + 2*a*d)*f^2)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(
1/3) + (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x
+ c)^(2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*
f + (b*d*e - a*d*f)*x))/(d*x + c)) - 2*(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e
*f^2)^(2/3)*(3*b*d*e^2 - (b*c + 2*a*d)*e*f + (3*b*d*e*f - (b*c + 2*a*d)*f^2)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c
+ a*d)*e*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c +
a^2*d)*e*f^2)^(2/3)*(d*x + c))/(d*x + c)) + 3*(b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c +
a^2*d)*e^2*f^2 + (b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(
1/3)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*d^2 - (-b*d^2)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*d + (-b*d^2)^(2
/3)*(d*x + c))/(d*x + c)) - 6*(b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2 +
(b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*log(((b*x +
a)^(1/3)*(d*x + c)^(2/3)*d + (-b*d^2)^(1/3)*(d*x + c))/(d*x + c)) + 6*(b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a
*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^2*d*e^4*f^2 - a^2*c*e*f^5 - (b^2*
c + 2*a*b*d)*e^3*f^3 + (2*a*b*c + a^2*d)*e^2*f^4 + (b^2*d*e^3*f^3 - a^2*c*f^6 - (b^2*c + 2*a*b*d)*e^2*f^4 + (2
*a*b*c + a^2*d)*e*f^5)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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