3.18.91 \(\int \frac {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx\) [1791]

Optimal. Leaf size=84 \[ \frac {6 b^2 (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {11}{6},\frac {19}{6};\frac {17}{6};-\frac {d (a+b x)}{b c-a d}\right )}{11 (b c-a d)^3 \sqrt [6]{c+d x}} \]

[Out]

6/11*b^2*(b*x+a)^(11/6)*(b*(d*x+c)/(-a*d+b*c))^(1/6)*hypergeom([11/6, 19/6],[17/6],-d*(b*x+a)/(-a*d+b*c))/(-a*
d+b*c)^3/(d*x+c)^(1/6)

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Rubi [A]
time = 0.01, antiderivative size = 84, 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 b^2 (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {11}{6},\frac {19}{6};\frac {17}{6};-\frac {d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/6)/(c + d*x)^(19/6),x]

[Out]

(6*b^2*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[11/6, 19/6, 17/6, -((d*(a + b*x))/
(b*c - a*d))])/(11*(b*c - a*d)^3*(c + d*x)^(1/6))

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 {(a+b x)^{5/6}}{(c+d x)^{19/6}} \, dx &=\frac {\left (b^3 \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 )^{19/6}} \, dx}{(b c-a d)^3 \sqrt [6]{c+d x}}\\ &=\frac {6 b^2 (a+b x)^{11/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {11}{6},\frac {19}{6};\frac {17}{6};-\frac {d (a+b x)}{b c-a d}\right )}{11 (b c-a d)^3 \sqrt [6]{c+d x}}\\ \end {align*}

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Mathematica [A]
time = 10.05, size = 81, normalized size = 0.96 \begin {gather*} \frac {6 b (a+b x)^{11/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (\frac {11}{6},\frac {19}{6};\frac {17}{6};\frac {d (a+b x)}{-b c+a d}\right )}{11 (b c-a d)^2 (c+d x)^{7/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/6)/(c + d*x)^(19/6),x]

[Out]

(6*b*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[11/6, 19/6, 17/6, (d*(a + b*x))/(-(b
*c) + a*d)])/(11*(b*c - a*d)^2*(c + d*x)^(7/6))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {5}{6}}}{\left (d x +c \right )^{\frac {19}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/6)/(d*x+c)^(19/6),x)

[Out]

int((b*x+a)^(5/6)/(d*x+c)^(19/6),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((b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(19/6), 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((b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="fricas")

[Out]

integral((b*x + a)^(5/6)*(d*x + c)^(5/6)/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/6)/(d*x+c)**(19/6),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5985 deep

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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((b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(19/6), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/6}}{{\left (c+d\,x\right )}^{19/6}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(5/6)/(c + d*x)^(19/6),x)

[Out]

int((a + b*x)^(5/6)/(c + d*x)^(19/6), x)

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