3.1788 \(\int \frac {(a+b x)^{5/6}}{\sqrt [6]{c+d x}} \, dx\)

Optimal. Leaf size=74 \[ \frac {6 (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{6},\frac {11}{6};\frac {17}{6};-\frac {d (a+b x)}{b c-a d}\right )}{11 b \sqrt [6]{c+d x}} \]

[Out]

6/11*(b*x+a)^(11/6)*(b*(d*x+c)/(-a*d+b*c))^(1/6)*hypergeom([1/6, 11/6],[17/6],-d*(b*x+a)/(-a*d+b*c))/b/(d*x+c)
^(1/6)

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Rubi [A]  time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {70, 69} \[ \frac {6 (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{6},\frac {11}{6};\frac {17}{6};-\frac {d (a+b x)}{b c-a d}\right )}{11 b \sqrt [6]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[1/6, 11/6, 17/6, -((d*(a + b*x))/(b*c
- a*d))])/(11*b*(c + d*x)^(1/6))

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rubi steps

\begin {align*} \int \frac {(a+b x)^{5/6}}{\sqrt [6]{c+d x}} \, dx &=\frac {\sqrt [6]{\frac {b (c+d x)}{b c-a d}} \int \frac {(a+b x)^{5/6}}{\sqrt [6]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{\sqrt [6]{c+d x}}\\ &=\frac {6 (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{6},\frac {11}{6};\frac {17}{6};-\frac {d (a+b x)}{b c-a d}\right )}{11 b \sqrt [6]{c+d x}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 73, normalized size = 0.99 \[ \frac {6 (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{6},\frac {11}{6};\frac {17}{6};\frac {d (a+b x)}{a d-b c}\right )}{11 b \sqrt [6]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[1/6, 11/6, 17/6, (d*(a + b*x))/(-(b*c)
+ a*d)])/(11*b*(c + d*x)^(1/6))

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fricas [F]  time = 1.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {1}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {5}{6}}}{\left (d x +c \right )^{\frac {1}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{\frac {5}{6}}}{{\left (d x + c\right )}^{\frac {1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^{5/6}}{{\left (c+d\,x\right )}^{1/6}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{\frac {5}{6}}}{\sqrt [6]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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