3.1834 \(\int \frac{\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx\)

Optimal. Leaf size=332 \[ -\frac{\sqrt [6]{d} \log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac{\sqrt [6]{d} \log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac{\sqrt{3} \sqrt [6]{d} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac{\sqrt{3} \sqrt [6]{d} \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{b^{7/6}}+\frac{2 \sqrt [6]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac{6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}} \]

[Out]

(-6*(c + d*x)^(1/6))/(b*(a + b*x)^(1/6)) + (Sqrt[3]*d^(1/6)*ArcTan[1/Sqrt[3] - (
2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/b^(7/6) - (Sqrt[3
]*d^(1/6)*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d
*x)^(1/6))])/b^(7/6) + (2*d^(1/6)*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c
+ d*x)^(1/6))])/b^(7/6) - (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^(7/6)) + (
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^(7/6))

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Rubi [A]  time = 0.912692, antiderivative size = 332, 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 \[ -\frac{\sqrt [6]{d} \log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac{\sqrt [6]{d} \log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac{\sqrt{3} \sqrt [6]{d} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac{\sqrt{3} \sqrt [6]{d} \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{b^{7/6}}+\frac{2 \sqrt [6]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac{6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(c + d*x)^(1/6))/(b*(a + b*x)^(1/6)) + (Sqrt[3]*d^(1/6)*ArcTan[1/Sqrt[3] - (
2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/b^(7/6) - (Sqrt[3
]*d^(1/6)*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d
*x)^(1/6))])/b^(7/6) + (2*d^(1/6)*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c
+ d*x)^(1/6))])/b^(7/6) - (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^(7/6)) + (
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^(7/6))

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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)**(7/6),x)

[Out]

Timed out

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Mathematica [C]  time = 0.102547, size = 74, normalized size = 0.22 \[ \frac{6 \sqrt [6]{c+d x} \left (\sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )-1\right )}{b \sqrt [6]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(6*(c + d*x)^(1/6)*(-1 + ((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*(a + b*x)^(1/6))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{1\sqrt [6]{dx+c} \left ( bx+a \right ) ^{-{\frac{7}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((d*x+c)^(1/6)/(b*x+a)^(7/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{7}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/6)/(b*x + a)^(7/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/6)/(b*x + a)^(7/6), x)

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Fricas [A]  time = 0.242968, size = 857, normalized size = 2.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/6)/(b*x + a)^(7/6),x, algorithm="fricas")

[Out]

-1/2*(4*sqrt(3)*(b^2*x + a*b)*(d/b^7)^(1/6)*arctan(sqrt(3)*(b^2*x + a*b)*(d/b^7)
^(1/6)/(2*(b*x + a)*sqrt(((b*x + a)^(5/6)*(d*x + c)^(1/6)*b*(d/b^7)^(1/6) + (b^3
*x + a*b^2)*(d/b^7)^(1/3) + (b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) + (b^2*x
+ a*b)*(d/b^7)^(1/6) + 2*(b*x + a)^(5/6)*(d*x + c)^(1/6))) + 4*sqrt(3)*(b^2*x +
a*b)*(d/b^7)^(1/6)*arctan(sqrt(3)*(b^2*x + a*b)*(d/b^7)^(1/6)/(2*(b*x + a)*sqrt
(-((b*x + a)^(5/6)*(d*x + c)^(1/6)*b*(d/b^7)^(1/6) - (b^3*x + a*b^2)*(d/b^7)^(1/
3) - (b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) - (b^2*x + a*b)*(d/b^7)^(1/6) +
2*(b*x + a)^(5/6)*(d*x + c)^(1/6))) - (b^2*x + a*b)*(d/b^7)^(1/6)*log(4*((b*x +
a)^(5/6)*(d*x + c)^(1/6)*b*(d/b^7)^(1/6) + (b^3*x + a*b^2)*(d/b^7)^(1/3) + (b*x
+ a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) + (b^2*x + a*b)*(d/b^7)^(1/6)*log(-4*((b
*x + a)^(5/6)*(d*x + c)^(1/6)*b*(d/b^7)^(1/6) - (b^3*x + a*b^2)*(d/b^7)^(1/3) -
(b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) - 2*(b^2*x + a*b)*(d/b^7)^(1/6)*log(
((b^2*x + a*b)*(d/b^7)^(1/6) + (b*x + a)^(5/6)*(d*x + c)^(1/6))/(b*x + a)) + 2*(
b^2*x + a*b)*(d/b^7)^(1/6)*log(-((b^2*x + a*b)*(d/b^7)^(1/6) - (b*x + a)^(5/6)*(
d*x + c)^(1/6))/(b*x + a)) + 12*(b*x + a)^(5/6)*(d*x + c)^(1/6))/(b^2*x + a*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [6]{c + d x}}{\left (a + b x\right )^{\frac{7}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/6)/(b*x+a)**(7/6),x)

[Out]

Integral((c + d*x)**(1/6)/(a + b*x)**(7/6), 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)^(7/6),x, algorithm="giac")

[Out]

Timed out