3.1827 \(\int \frac{1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx\)

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

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

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Rubi [A]  time = 0.726441, 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 \[ -\frac{\log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{5/6} \sqrt [6]{d}}+\frac{\log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{5/6} \sqrt [6]{d}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}} \]

Antiderivative was successfully verified.

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

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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)**(5/6)/(d*x+c)**(1/6),x)

[Out]

Timed out

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Mathematica [C]  time = 0.0646161, size = 73, normalized size = 0.24 \[ \frac{6 (c+d x)^{5/6} \left (\frac{d (a+b x)}{a d-b c}\right )^{5/6} \, _2F_1\left (\frac{5}{6},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )}{5 d (a+b x)^{5/6}} \]

Antiderivative was successfully verified.

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

[Out]

(6*((d*(a + b*x))/(-(b*c) + a*d))^(5/6)*(c + d*x)^(5/6)*Hypergeometric2F1[5/6, 5
/6, 11/6, (b*(c + d*x))/(b*c - a*d)])/(5*d*(a + b*x)^(5/6))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{6}}}{\frac{1}{\sqrt [6]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244462, 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)^(5/6)*(d*x + c)^(1/6)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out