3.1807 \(\int \frac{\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx\)
Optimal. Leaf size=378 \[ -\frac{(b c-a d) \log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{7/6} d^{5/6}}+\frac{(b c-a d) \log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{7/6} d^{5/6}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt{3} b^{7/6} d^{5/6}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt{3} b^{7/6} d^{5/6}}+\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{7/6} d^{5/6}}+\frac{(a+b x)^{5/6} \sqrt [6]{c+d x}}{b} \]
[Out]
((a + b*x)^(5/6)*(c + d*x)^(1/6))/b + ((b*c - a*d)*ArcTan[1/Sqrt[3] - (2*d^(1/6)
*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/(2*Sqrt[3]*b^(7/6)*d^(5/6)
) - ((b*c - a*d)*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)
*(c + d*x)^(1/6))])/(2*Sqrt[3]*b^(7/6)*d^(5/6)) + ((b*c - a*d)*ArcTanh[(d^(1/6)*
(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(3*b^(7/6)*d^(5/6)) - ((b*c - a*d)*
Log[b^(1/3) + (d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3) - (b^(1/6)*d^(1/6)*(a +
b*x)^(1/6))/(c + d*x)^(1/6)])/(12*b^(7/6)*d^(5/6)) + ((b*c - a*d)*Log[b^(1/3) +
(d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3) + (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(c
+ d*x)^(1/6)])/(12*b^(7/6)*d^(5/6))
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Rubi [A] time = 0.922887, antiderivative size = 378, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421
\[ -\frac{(b c-a d) \log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{7/6} d^{5/6}}+\frac{(b c-a d) \log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{7/6} d^{5/6}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt{3} b^{7/6} d^{5/6}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt{3} b^{7/6} d^{5/6}}+\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{7/6} d^{5/6}}+\frac{(a+b x)^{5/6} \sqrt [6]{c+d x}}{b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(1/6)/(a + b*x)^(1/6),x]
[Out]
((a + b*x)^(5/6)*(c + d*x)^(1/6))/b + ((b*c - a*d)*ArcTan[1/Sqrt[3] - (2*d^(1/6)
*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/(2*Sqrt[3]*b^(7/6)*d^(5/6)
) - ((b*c - a*d)*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)
*(c + d*x)^(1/6))])/(2*Sqrt[3]*b^(7/6)*d^(5/6)) + ((b*c - a*d)*ArcTanh[(d^(1/6)*
(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(3*b^(7/6)*d^(5/6)) - ((b*c - a*d)*
Log[b^(1/3) + (d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3) - (b^(1/6)*d^(1/6)*(a +
b*x)^(1/6))/(c + d*x)^(1/6)])/(12*b^(7/6)*d^(5/6)) + ((b*c - a*d)*Log[b^(1/3) +
(d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3) + (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(c
+ d*x)^(1/6)])/(12*b^(7/6)*d^(5/6))
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/6)/(b*x+a)**(1/6),x)
[Out]
Timed out
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Mathematica [C] time = 0.153086, size = 90, normalized size = 0.24 \[ \frac{\sqrt [6]{c+d x} \left ((b c-a d) \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x)\right )}{b d \sqrt [6]{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(1/6)/(a + b*x)^(1/6),x]
[Out]
((c + d*x)^(1/6)*(d*(a + b*x) + (b*c - a*d)*((d*(a + b*x))/(-(b*c) + a*d))^(1/6)
*Hypergeometric2F1[1/6, 1/6, 7/6, (b*(c + d*x))/(b*c - a*d)]))/(b*d*(a + b*x)^(1
/6))
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Maple [F] time = 0.031, size = 0, normalized size = 0. \[ \int{1\sqrt [6]{dx+c}{\frac{1}{\sqrt [6]{bx+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/6)/(b*x+a)^(1/6),x)
[Out]
int((d*x+c)^(1/6)/(b*x+a)^(1/6),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{1}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(1/6)/(b*x + a)^(1/6),x, algorithm="maxima")
[Out]
integrate((d*x + c)^(1/6)/(b*x + a)^(1/6), x)
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Fricas [A] time = 0.286854, size = 3193, normalized size = 8.45 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(1/6)/(b*x + a)^(1/6),x, algorithm="fricas")
[Out]
1/12*(4*sqrt(3)*b*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^
3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)*arctan(-s
qrt(3)*(b^2*d*x + a*b*d)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3
*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)/(2
*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - 2*(b*x + a)*sqrt(((b^2*c*d - a*b*
d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*
d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^
5))^(1/6) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b
^3*d^2*x + a*b^2*d^2)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^
3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/3))/(b*x
+ a)) + (b^2*d*x + a*b*d)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a
^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6))
) + 4*sqrt(3)*b*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*
d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)*arctan(-sqr
t(3)*(b^2*d*x + a*b*d)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b
^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)/(2*(
b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - 2*(b*x + a)*sqrt(-((b^2*c*d - a*b*d
^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d
^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5
))^(1/6) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (b^
3*d^2*x + a*b^2*d^2)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3
*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/3))/(b*x
+ a)) - (b^2*d*x + a*b*d)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^
3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)))
+ b*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^
4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)*log(((b^2*c*d - a*b*d^
2)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^
2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5)
)^(1/6) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b^3
*d^2*x + a*b^2*d^2)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*
c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/3))/(b*x +
a)) - b*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 1
5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)*log(-((b^2*c*d - a
*b*d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c
^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7
*d^5))^(1/6) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) -
(b^3*d^2*x + a*b^2*d^2)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3
*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/3))/(
b*x + a)) + 2*b*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*
d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)*log(-((b*c
- a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^2*d*x + a*b*d)*((b^6*c^6 - 6*a*b^5*c
^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*
d^5 + a^6*d^6)/(b^7*d^5))^(1/6))/(b*x + a)) - 2*b*((b^6*c^6 - 6*a*b^5*c^5*d + 15
*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6
*d^6)/(b^7*d^5))^(1/6)*log(-((b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - (b^2*
d*x + a*b*d)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3
+ 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6))/(b*x + a)) +
12*(b*x + a)^(5/6)*(d*x + c)^(1/6))/b
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [6]{c + d x}}{\sqrt [6]{a + b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/6)/(b*x+a)**(1/6),x)
[Out]
Integral((c + d*x)**(1/6)/(a + b*x)**(1/6), x)
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(1/6)/(b*x + a)^(1/6),x, algorithm="giac")
[Out]
Timed out