3.1808 \(\int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx\)

Optimal. Leaf size=309 \[ -\frac{\log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac{\log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}} \]

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*
x)^(1/6))])/(b^(1/6)*d^(5/6)) - (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)
^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/(b^(1/6)*d^(5/6)) + (2*ArcTanh[(d^(1
/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(b^(1/6)*d^(5/6)) - 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)]/(2*b^(1/6)*d^(5/6)) + 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)]/(2*b^(1/6)*
d^(5/6))

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Rubi [A]  time = 0.856433, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{\log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac{\log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(1/6)*(c + d*x)^(5/6)),x]

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*
x)^(1/6))])/(b^(1/6)*d^(5/6)) - (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)
^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/(b^(1/6)*d^(5/6)) + (2*ArcTanh[(d^(1
/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(b^(1/6)*d^(5/6)) - 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)]/(2*b^(1/6)*d^(5/6)) + 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)]/(2*b^(1/6)*
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(1/(b*x+a)**(1/6)/(d*x+c)**(5/6),x)

[Out]

Timed out

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Mathematica [C]  time = 0.0663434, size = 71, normalized size = 0.23 \[ \frac{6 \sqrt [6]{c+d x} \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )}{d \sqrt [6]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(5/6)),x]

[Out]

(6*((d*(a + b*x))/(-(b*c) + a*d))^(1/6)*(c + d*x)^(1/6)*Hypergeometric2F1[1/6, 1
/6, 7/6, (b*(c + d*x))/(b*c - a*d)])/(d*(a + b*x)^(1/6))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [6]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{5}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/6)/(d*x+c)^(5/6),x)

[Out]

int(1/(b*x+a)^(1/6)/(d*x+c)^(5/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(5/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(5/6)), x)

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Fricas [A]  time = 0.256867, size = 797, normalized size = 2.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(5/6)),x, algorithm="fricas")

[Out]

-2*sqrt(3)*(1/(b*d^5))^(1/6)*arctan(sqrt(3)*(b*d*x + a*d)*(1/(b*d^5))^(1/6)/(2*(
b*x + a)*sqrt(((b*x + a)^(5/6)*(d*x + c)^(1/6)*d*(1/(b*d^5))^(1/6) + (b*d^2*x +
a*d^2)*(1/(b*d^5))^(1/3) + (b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) + (b*d*x
+ a*d)*(1/(b*d^5))^(1/6) + 2*(b*x + a)^(5/6)*(d*x + c)^(1/6))) - 2*sqrt(3)*(1/(b
*d^5))^(1/6)*arctan(sqrt(3)*(b*d*x + a*d)*(1/(b*d^5))^(1/6)/(2*(b*x + a)*sqrt(-(
(b*x + a)^(5/6)*(d*x + c)^(1/6)*d*(1/(b*d^5))^(1/6) - (b*d^2*x + a*d^2)*(1/(b*d^
5))^(1/3) - (b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) - (b*d*x + a*d)*(1/(b*d^
5))^(1/6) + 2*(b*x + a)^(5/6)*(d*x + c)^(1/6))) + 1/2*(1/(b*d^5))^(1/6)*log(4*((
b*x + a)^(5/6)*(d*x + c)^(1/6)*d*(1/(b*d^5))^(1/6) + (b*d^2*x + a*d^2)*(1/(b*d^5
))^(1/3) + (b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) - 1/2*(1/(b*d^5))^(1/6)*l
og(-4*((b*x + a)^(5/6)*(d*x + c)^(1/6)*d*(1/(b*d^5))^(1/6) - (b*d^2*x + a*d^2)*(
1/(b*d^5))^(1/3) - (b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) + (1/(b*d^5))^(1/
6)*log(((b*d*x + a*d)*(1/(b*d^5))^(1/6) + (b*x + a)^(5/6)*(d*x + c)^(1/6))/(b*x
+ a)) - (1/(b*d^5))^(1/6)*log(-((b*d*x + a*d)*(1/(b*d^5))^(1/6) - (b*x + a)^(5/6
)*(d*x + c)^(1/6))/(b*x + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [6]{a + b x} \left (c + d x\right )^{\frac{5}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(1/6)/(d*x+c)**(5/6),x)

[Out]

Integral(1/((a + b*x)**(1/6)*(c + d*x)**(5/6)), 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(1/((b*x + a)^(1/6)*(d*x + c)^(5/6)),x, algorithm="giac")

[Out]

Timed out