3.31.6 \(\int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^3} \, dx\) [3006]
Optimal. Leaf size=325 \[ \frac {\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (d e-c f) (e+f x)^2}-\frac {(b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{6 (b e-a f) (d e-c f) (e+f x)}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{4/3}}-\frac {(b c-a d)^2 \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{4/3}}+\frac {(b c-a d)^2 \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{4/3}} \]
[Out]
1/2*(b*x+a)^(1/3)*(d*x+c)^(5/3)/(-c*f+d*e)/(f*x+e)^2-1/6*(-a*d+b*c)*(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-a*f+b*e)/(-c
*f+d*e)/(f*x+e)-1/18*(-a*d+b*c)^2*ln(f*x+e)/(-a*f+b*e)^(5/3)/(-c*f+d*e)^(4/3)+1/6*(-a*d+b*c)^2*ln(-(b*x+a)^(1/
3)+(-a*f+b*e)^(1/3)*(d*x+c)^(1/3)/(-c*f+d*e)^(1/3))/(-a*f+b*e)^(5/3)/(-c*f+d*e)^(4/3)+1/9*(-a*d+b*c)^2*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))/(-a*f+b*e)^(5/3)/(-c*f+
d*e)^(4/3)*3^(1/2)
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Rubi [A]
time = 0.16, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {96, 93}
\begin {gather*} \frac {(b c-a d)^2 \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 )}{3 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{4/3}}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{6 (e+f x) (b e-a f) (d e-c f)}+\frac {\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (e+f x)^2 (d e-c f)}-\frac {(b c-a d)^2 \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{4/3}}+\frac {(b c-a d)^2 \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 )}{6 (b e-a f)^{5/3} (d e-c f)^{4/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^(1/3)*(c + d*x)^(2/3))/(e + f*x)^3,x]
[Out]
((a + b*x)^(1/3)*(c + d*x)^(5/3))/(2*(d*e - c*f)*(e + f*x)^2) - ((b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/
(6*(b*e - a*f)*(d*e - c*f)*(e + f*x)) + ((b*c - a*d)^2*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))])/(3*Sqrt[3]*(b*e - a*f)^(5/3)*(d*e - c*f)^(4/3)) - ((b*c - a*d)
^2*Log[e + f*x])/(18*(b*e - a*f)^(5/3)*(d*e - c*f)^(4/3)) + ((b*c - a*d)^2*Log[-(a + b*x)^(1/3) + ((b*e - a*f)
^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(6*(b*e - a*f)^(5/3)*(d*e - c*f)^(4/3))
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 96
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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^3} \, dx &=\frac {\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (d e-c f) (e+f x)^2}-\frac {(b c-a d) \int \frac {(c+d x)^{2/3}}{(a+b x)^{2/3} (e+f x)^2} \, dx}{6 (d e-c f)}\\ &=\frac {\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (d e-c f) (e+f x)^2}-\frac {(b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{6 (b e-a f) (d e-c f) (e+f x)}-\frac {(b c-a d)^2 \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{9 (b e-a f) (d e-c f)}\\ &=\frac {\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (d e-c f) (e+f x)^2}-\frac {(b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{6 (b e-a f) (d e-c f) (e+f x)}+\frac {(b c-a d)^2 \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 )}{3 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{4/3}}-\frac {(b c-a d)^2 \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{4/3}}+\frac {(b c-a d)^2 \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 )}{6 (b e-a f)^{5/3} (d e-c f)^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 3.45, size = 375, normalized size = 1.15 \begin {gather*} \frac {1}{18} (b c-a d)^2 \left (\frac {3 \sqrt [3]{a+b x} (c+d x)^{2/3} (b (-2 c e-3 d e x+c f x)+a (-d e+3 c f+2 d f x))}{(b c-a d)^2 (b e-a f) (-d e+c f) (e+f x)^2}-\frac {2 \sqrt {3} \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)^{5/3} (-d e+c f)^{4/3}}+\frac {2 \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)^{5/3} (-d e+c f)^{4/3}}-\frac {\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)^{5/3} (-d e+c f)^{4/3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^(1/3)*(c + d*x)^(2/3))/(e + f*x)^3,x]
[Out]
((b*c - a*d)^2*((3*(a + b*x)^(1/3)*(c + d*x)^(2/3)*(b*(-2*c*e - 3*d*e*x + c*f*x) + a*(-(d*e) + 3*c*f + 2*d*f*x
)))/((b*c - a*d)^2*(b*e - a*f)*(-(d*e) + c*f)*(e + f*x)^2) - (2*Sqrt[3]*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)^(5/3)*(-(d*e) + c*f)^(4/3)) + (2*Lo
g[(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)^(5/3)*(-(d*e) + c*
f)^(4/3)) - 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)^(5/3)*(-(d*e) + c*f)^(4/3))))/18
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}{\left (f x +e \right )^{3}}\, dx\]
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)^3,x)
[Out]
int((b*x+a)^(1/3)*(d*x+c)^(2/3)/(f*x+e)^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((b*x+a)^(1/3)*(d*x+c)^(2/3)/(f*x+e)^3,x, algorithm="maxima")
[Out]
integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)/(f*x + e)^3, x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1902 vs.
\(2 (291) = 582\).
time = 1.78, size = 3958, normalized size = 12.18 \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)^3,x, algorithm="fricas")
[Out]
[-1/18*(3*sqrt(1/3)*((a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^4*x^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)
*e^4 + (2*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*f*x - (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*f)*e^3
+ ((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*f^2*x^2 - 2*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*f^2*x
+ (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^2)*e^2 - ((b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*f^3*x^2
- 2*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^3*x)*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)) - (-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^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*x^2 + 2*(b^2
*c^2 - 2*a*b*c*d + a^2*d^2)*f*x*e + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2)*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^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*x^2 + 2*(b^2*c^2 - 2*a*b*c*d
+ a^2*d^2)*f*x*e + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2)*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*(3*a^3*c^2*f^4 + (a^2*b*c^2 + 2*a^3*c*d)*f^4*x + (3*b^3*d^2*x + 2*b^3*c*d + a*b^2*d^2)*
e^4 - 2*(2*(b^3*c*d + 2*a*b^2*d^2)*f*x + (b^3*c^2 + 4*a*b^2*c*d + a^2*b*d^2)*f)*e^3 + ((b^3*c^2 + 10*a*b^2*c*d
+ 7*a^2*b*d^2)*f^2*x + (7*a*b^2*c^2 + 10*a^2*b*c*d + a^3*d^2)*f^2)*e^2 - 2*((a*b^2*c^2 + 4*a^2*b*c*d + a^3*d^
2)*f^3*x + 2*(2*a^2*b*c^2 + a^3*c*d)*f^3)*e)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(a^3*c^2*f^7*x^2 - b^3*d^2*e^7 -
(2*b^3*d^2*f*x - (2*b^3*c*d + 3*a*b^2*d^2)*f)*e^6 - (b^3*d^2*f^2*x^2 - 2*(2*b^3*c*d + 3*a*b^2*d^2)*f^2*x + (b
^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f^2)*e^5 + ((2*b^3*c*d + 3*a*b^2*d^2)*f^3*x^2 - 2*(b^3*c^2 + 6*a*b^2*c*d +
3*a^2*b*d^2)*f^3*x + (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^3)*e^4 - ((b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*
f^4*x^2 - 2*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^4*x + (3*a^2*b*c^2 + 2*a^3*c*d)*f^4)*e^3 + (a^3*c^2*f^5 +
(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^5*x^2 - 2*(3*a^2*b*c^2 + 2*a^3*c*d)*f^5*x)*e^2 + (2*a^3*c^2*f^6*x - (3
*a^2*b*c^2 + 2*a^3*c*d)*f^6*x^2)*e), 1/18*(6*sqrt(1/3)*((a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^4*x^2 + (b^3
*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^4 + (2*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*f*x - (b^3*c^3 - a*b^2*c^
2*d - a^2*b*c*d^2 + a^3*d^3)*f)*e^3 + ((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*f^2*x^2 - 2*(b^3*c^3 - a*b^2*c^
2*d - a^2*b*c*d^2 + a^3*d^3)*f^2*x + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^2)*e^2 - ((b^3*c^3 - a*b^2*c^2*
d - a^2*b*c*d^2 + a^3*d^3)*f^3*x^2 - 2*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^3*x)*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)) + (-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^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*x^2 + 2*(b^2*c^2 -
2*a*b*c*d + a^2*d^2)*f*x*e + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2)*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^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*x^2 + 2*(b^2*c^2 - 2*a*b*c*d + a^2*
d^2)*f*x*e + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^...
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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 )^{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)/(f*x+e)**3,x)
[Out]
Integral((a + b*x)**(1/3)*(c + d*x)**(2/3)/(e + f*x)**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((b*x+a)^(1/3)*(d*x+c)^(2/3)/(f*x+e)^3,x, algorithm="giac")
[Out]
integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)/(f*x + e)^3, 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 )}^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))/(e + f*x)^3,x)
[Out]
int(((a + b*x)^(1/3)*(c + d*x)^(2/3))/(e + f*x)^3, x)
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