3.19.39 \(\int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx\) [1839]

Optimal. Leaf size=72 \[ -\frac {6 (c+d x)^{5/6} \, _2F_1\left (-\frac {5}{6},-\frac {1}{6};\frac {5}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 72, 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 = {72, 71} \begin {gather*} -\frac {6 (c+d x)^{5/6} \, _2F_1\left (-\frac {5}{6},-\frac {1}{6};\frac {5}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(c + d*x)^(5/6)*Hypergeometric2F1[-5/6, -1/6, 5/6, -((d*(a + b*x))/(b*c - a*d))])/(b*(a + b*x)^(1/6)*((b*(
c + d*x))/(b*c - a*d))^(5/6))

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 72

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 {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx &=\frac {(c+d x)^{5/6} \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{5/6}}{(a+b x)^{7/6}} \, dx}{\left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}}\\ &=-\frac {6 (c+d x)^{5/6} \, _2F_1\left (-\frac {5}{6},-\frac {1}{6};\frac {5}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}}\\ \end {align*}

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Mathematica [A]
time = 10.02, size = 71, normalized size = 0.99 \begin {gather*} -\frac {6 (c+d x)^{5/6} \, _2F_1\left (-\frac {5}{6},-\frac {1}{6};\frac {5}{6};\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(c + d*x)^(5/6)*Hypergeometric2F1[-5/6, -1/6, 5/6, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(a + b*x)^(1/6)*((b*(
c + d*x))/(b*c - a*d))^(5/6))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {5}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {5}{6}}}{\left (a + b x\right )^{\frac {7}{6}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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