3.26.16 \(\int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx\)
Optimal. Leaf size=197 \[ \frac {\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt {3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}} \]
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Rubi [A] time = 0.07, antiderivative size = 197, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used =
{91} \begin {gather*} \frac {\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt {3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)),x]
[Out]
-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(d*e - c*f)^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*(b*e - a*f)^(1/3)*(c + d*x)^(1/3)
)])/((b*e - a*f)^(1/3)*(d*e - c*f)^(2/3))) + Log[e + f*x]/(2*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)) - (3*Log[((d
*e - c*f)^(1/3)*(a + b*x)^(1/3))/(b*e - a*f)^(1/3) - (c + d*x)^(1/3)])/(2*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3))
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]
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}}+\frac {\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 71, normalized size = 0.36 \begin {gather*} \frac {3 (a+b x)^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 (c+d x)^{2/3} (b e-a f)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)),x]
[Out]
(3*(a + b*x)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/(2*(b*e -
a*f)*(c + d*x)^(2/3))
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IntegrateAlgebraic [A] time = 0.39, size = 275, normalized size = 1.40 \begin {gather*} \frac {\log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{a f-b e}}{\sqrt [3]{a+b x}}+\sqrt [3]{d e-c f}\right )}{\sqrt [3]{a f-b e} (d e-c f)^{2/3}}-\frac {\log \left (-\frac {\sqrt [3]{c+d x} \sqrt [3]{a f-b e} \sqrt [3]{d e-c f}}{\sqrt [3]{a+b x}}+\frac {(c+d x)^{2/3} (a f-b e)^{2/3}}{(a+b x)^{2/3}}+(d e-c f)^{2/3}\right )}{2 \sqrt [3]{a f-b e} (d e-c f)^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c+d x} \sqrt [3]{a f-b e}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}\right )}{\sqrt [3]{a f-b e} (d e-c f)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)),x]
[Out]
-((Sqrt[3]*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))])/((-(b*e) + a*f)^(1/3)*(d*e - c*f)^(2/3))) + Log[(d*e - c*f)^(1/3) + ((-(b*e) + a*f)^(1/3)*(c + d*x)^(1/
3))/(a + b*x)^(1/3)]/((-(b*e) + a*f)^(1/3)*(d*e - c*f)^(2/3)) - Log[(d*e - c*f)^(2/3) - ((-(b*e) + a*f)^(1/3)*
(d*e - c*f)^(1/3)*(c + d*x)^(1/3))/(a + b*x)^(1/3) + ((-(b*e) + a*f)^(2/3)*(c + d*x)^(2/3))/(a + b*x)^(2/3)]/(
2*(-(b*e) + a*f)^(1/3)*(d*e - c*f)^(2/3))
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fricas [B] time = 1.27, size = 1825, normalized size = 9.26
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e),x, algorithm="fricas")
[Out]
[-1/2*(sqrt(3)*(b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*sqrt((-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f -
(b*c^2 + 2*a*c*d)*e*f^2)^(1/3)/(b*e - a*f))*log((3*a*c^2*f^2 + (2*b*c*d + a*d^2)*e^2 - 2*(b*c^2 + 2*a*c*d)*e*f
+ 3*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(d*e - c*f)*(b*x + a)^
(2/3)*(d*x + c)^(1/3) + (3*b*d^2*e^2 - 2*(2*b*c*d + a*d^2)*e*f + (b*c^2 + 2*a*c*d)*f^2)*x - sqrt(3)*(2*(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*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e
^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a
*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(a*d*e - a*c*f + (b*d*e - b*c*f)*x))*sqrt((-b*d^2*e^3 + a*c^2*f^3
+ (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)/(b*e - a*f)))/(f*x + e)) + 2*(-b*d^2*e^3 + a*c^2*f
^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*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*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3
)*(b*x + a))/(b*x + a)) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*l
og(((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*d^2*e^3 + a*c^2*f^3 + (2*b*c*d
+ a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (
2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(a*d*e - a*c*f + (b*d*e - b*c*f)*x))/(b*x + a)))/(b*d^
2*e^3 - a*c^2*f^3 - (2*b*c*d + a*d^2)*e^2*f + (b*c^2 + 2*a*c*d)*e*f^2), 1/2*(2*sqrt(3)*(b*d*e^2 + a*c*f^2 - (b
*c + a*d)*e*f)*sqrt(-(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)/(b*e -
a*f))*arctan(1/3*sqrt(3)*(2*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3
)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^
2)^(1/3)*(a*d*e - a*c*f + (b*d*e - b*c*f)*x))*sqrt(-(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2
+ 2*a*c*d)*e*f^2)^(1/3)/(b*e - a*f))/(a*d^2*e^2 - 2*a*c*d*e*f + a*c^2*f^2 + (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*
f^2)*x)) - 2*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*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*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2
*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a))/(b*x + a)) + (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f
- (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*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*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(
1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(a*d*e - a*c*f + (b*
d*e - b*c*f)*x))/(b*x + a)))/(b*d^2*e^3 - a*c^2*f^3 - (2*b*c*d + a*d^2)*e^2*f + (b*c^2 + 2*a*c*d)*e*f^2)]
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} {\left (f x + e\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e),x, algorithm="giac")
[Out]
integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)), x)
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (f x +e \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e),x)
[Out]
int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} {\left (f x + e\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e),x, algorithm="maxima")
[Out]
integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (e+f\,x\right )\,{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((e + f*x)*(a + b*x)^(1/3)*(c + d*x)^(2/3)),x)
[Out]
int(1/((e + f*x)*(a + b*x)^(1/3)*(c + d*x)^(2/3)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}} \left (e + f x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)**(1/3)/(d*x+c)**(2/3)/(f*x+e),x)
[Out]
Integral(1/((a + b*x)**(1/3)*(c + d*x)**(2/3)*(e + f*x)), x)
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