3.3012 \(\int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx\)
Optimal. Leaf size=256 \[ \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \log (e+f x)}{6 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac {(b c-a d) \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac {(b c-a d) \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} (b e-a f)^{2/3} (d e-c f)^{4/3}} \]
[Out]
(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-c*f+d*e)/(f*x+e)-1/6*(-a*d+b*c)*ln(f*x+e)/(-a*f+b*e)^(2/3)/(-c*f+d*e)^(4/3)+1/2*
(-a*d+b*c)*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)^(2/3)/(-c*f+d*e)^(4/3
)+1/3*(-a*d+b*c)*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)^(2/3)/(-c*f+d*e)^(4/3)*3^(1/2)
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Rubi [A] time = 0.08, antiderivative size = 256, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used =
{94, 91} \[ \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \log (e+f x)}{6 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac {(b c-a d) \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac {(b c-a d) \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} (b e-a f)^{2/3} (d e-c f)^{4/3}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^2),x]
[Out]
((a + b*x)^(1/3)*(c + d*x)^(2/3))/((d*e - c*f)*(e + f*x)) + ((b*c - a*d)*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]*(b*e - a*f)^(2/3)*(d*e - c*f)^(4/3)
) - ((b*c - a*d)*Log[e + f*x])/(6*(b*e - a*f)^(2/3)*(d*e - c*f)^(4/3)) + ((b*c - a*d)*Log[-(a + b*x)^(1/3) + (
(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(2*(b*e - a*f)^(2/3)*(d*e - c*f)^(4/3))
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]
Rule 94
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[p, 1] && !SumSimplerQ[m, 1])
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx &=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f) (e+f x)}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{3 (d e-c f)}\\ &=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f) (e+f x)}+\frac {(b c-a d) \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} (b e-a f)^{2/3} (d e-c f)^{4/3}}-\frac {(b c-a d) \log (e+f x)}{6 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac {(b c-a d) \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 (b e-a f)^{2/3} (d e-c f)^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 115, normalized size = 0.45 \[ \frac {\sqrt [3]{a+b x} \left ((c+d x) (b e-a f)-(e+f x) (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{\sqrt [3]{c+d x} (e+f x) (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^2),x]
[Out]
((a + b*x)^(1/3)*((b*e - a*f)*(c + d*x) - (b*c - a*d)*(e + f*x)*Hypergeometric2F1[1/3, 1, 4/3, ((d*e - c*f)*(a
+ b*x))/((b*e - a*f)*(c + d*x))]))/((b*e - a*f)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x))
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fricas [B] time = 1.08, size = 2504, normalized size = 9.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x, algorithm="fricas")
[Out]
[1/6*(3*sqrt(1/3)*((b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 - a^2*d^2)*e^2*f + (a*b*c^2 - a^2*c*d)*e*f^2 + ((b^2*c*d
- a*b*d^2)*e^2*f - (b^2*c^2 - a^2*d^2)*e*f^2 + (a*b*c^2 - a^2*c*d)*f^3)*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))*sqr
t(-(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))
- (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*c - a*d)*f*x + (b*c -
a*d)*e)*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)*((b*c - a*d)*f*x + (b*c
- a*d)*e)*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)) + 6*(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)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^2*d^2*e^5 + a^2*c^2*e*
f^4 - 2*(b^2*c*d + a*b*d^2)*e^4*f + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^2 - 2*(a*b*c^2 + a^2*c*d)*e^2*f^3 +
(b^2*d^2*e^4*f + a^2*c^2*f^5 - 2*(b^2*c*d + a*b*d^2)*e^3*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^3 - 2*(a*
b*c^2 + a^2*c*d)*e*f^4)*x), -1/6*(6*sqrt(1/3)*((b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 - a^2*d^2)*e^2*f + (a*b*c^2
- a^2*c*d)*e*f^2 + ((b^2*c*d - a*b*d^2)*e^2*f - (b^2*c^2 - a^2*d^2)*e*f^2 + (a*b*c^2 - a^2*c*d)*f^3)*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)) + (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*c - a*d)*f*x + (b*c - a*d)*e)*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)*((b*c - a*d)*f*x + (b*c - a*d)*e)*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)) - 6*(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)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^2*d^2*e^5 + a^2*c^2*e*f^4 - 2*(b^2*c*d + a
*b*d^2)*e^4*f + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^2 - 2*(a*b*c^2 + a^2*c*d)*e^2*f^3 + (b^2*d^2*e^4*f + a^2
*c^2*f^5 - 2*(b^2*c*d + a*b*d^2)*e^3*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^3 - 2*(a*b*c^2 + a^2*c*d)*e*f
^4)*x)]
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x, algorithm="giac")
[Out]
integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^2), x)
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} \left (f x +e \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x)
[Out]
int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x, algorithm="maxima")
[Out]
integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^(1/3)/((e + f*x)^2*(c + d*x)^(1/3)),x)
[Out]
int((a + b*x)^(1/3)/((e + f*x)^2*(c + d*x)^(1/3)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a + b x}}{\sqrt [3]{c + d x} \left (e + f x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**(1/3)/(d*x+c)**(1/3)/(f*x+e)**2,x)
[Out]
Integral((a + b*x)**(1/3)/((c + d*x)**(1/3)*(e + f*x)**2), x)
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