3.1789 \(\int \frac{(a+b x)^{5/6}}{(c+d x)^{7/6}} \, dx\)

Optimal. Leaf size=81 \[ \frac{6 (a+b x)^{11/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{7}{6},\frac{11}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)} \]

[Out]

(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[7/6, 11/6, 17/6, -((d*(a + b*x))/(b*c
- a*d))])/(11*(b*c - a*d)*(c + d*x)^(1/6))

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Rubi [A]  time = 0.0203077, antiderivative size = 81, normalized size of antiderivative = 1., 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} \[ \frac{6 (a+b x)^{11/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{7}{6},\frac{11}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/6)/(c + d*x)^(7/6),x]

[Out]

(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[7/6, 11/6, 17/6, -((d*(a + b*x))/(b*c
- a*d))])/(11*(b*c - a*d)*(c + d*x)^(1/6))

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])

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]))

Rubi steps

\begin{align*} \int \frac{(a+b x)^{5/6}}{(c+d x)^{7/6}} \, dx &=\frac{\left (b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}\right ) \int \frac{(a+b x)^{5/6}}{\left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{7/6}} \, dx}{(b c-a d) \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{7}{6},\frac{11}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 (b c-a d) \sqrt [6]{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0402007, size = 73, normalized size = 0.9 \[ \frac{6 (a+b x)^{11/6} \left (\frac{b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (\frac{7}{6},\frac{11}{6};\frac{17}{6};\frac{d (a+b x)}{a d-b c}\right )}{11 b (c+d x)^{7/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/6)/(c + d*x)^(7/6),x]

[Out]

(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[7/6, 11/6, 17/6, (d*(a + b*x))/(-(b*c)
+ a*d)])/(11*b*(c + d*x)^(7/6))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{7}{6}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/6)/(d*x+c)^(7/6),x)

[Out]

int((b*x+a)^(5/6)/(d*x+c)^(7/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{7}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/6)/(d*x+c)^(7/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(7/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/6)/(d*x+c)^(7/6),x, algorithm="fricas")

[Out]

integral((b*x + a)^(5/6)*(d*x + c)^(5/6)/(d^2*x^2 + 2*c*d*x + c^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/6)/(d*x+c)**(7/6),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{7}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/6)/(d*x+c)^(7/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(7/6), x)