3.31.5 \(\int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx\) [3005]

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

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 417, normalized size of antiderivative = 1.00, 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 = {99, 163, 61, 93} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{b} d^{2/3} \text {ArcTan}\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 {(-2 a d f-b c f+3 b d e) \text {ArcTan}\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 {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 {\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 {\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} \end {gather*}

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 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 93

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]

Rule 99

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 163

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]

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*}

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Mathematica [A]
time = 1.89, size = 524, normalized size = 1.26 \begin {gather*} \frac {-\frac {6 f \sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x}+6 \sqrt {3} \sqrt [3]{b} d^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )+\frac {2 \sqrt {3} (3 b d e-b c f-2 a d f) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{-d e+c f}}-6 \sqrt [3]{b} d^{2/3} \log \left (\sqrt [3]{b}-\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )+3 \sqrt [3]{b} d^{2/3} \log \left (b^{2/3}+\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}}\right )+\frac {2 (-3 b d e+b c f+2 a d f) \log \left (\sqrt [3]{b e-a f}+\frac {\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{(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 ((b e-a f)^{2/3}+\frac {(-d e+c f)^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}-\frac {\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{(b e-a f)^{2/3} \sqrt [3]{-d e+c f}}}{6 f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}{\left (f x +e \right )^{2}}\, dx\]

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 = 0.00, 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((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] Leaf count of result is larger than twice the leaf count of optimal. 1712 vs. \(2 (354) = 708\).
time = 5.44, size = 3578, normalized size = 8.58 \begin {gather*} \text {Too large to display} \end {gather*}

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

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

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

[Out]

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

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