3.19.15 \(\int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{7/6}} \, dx\) [1815]

Optimal. Leaf size=81 \[ \frac {6 (a+b x)^{5/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {11}{6};-\frac {d (a+b x)}{b c-a d}\right )}{5 (b c-a d) \sqrt [6]{c+d x}} \]

[Out]

6/5*(b*x+a)^(5/6)*(b*(d*x+c)/(-a*d+b*c))^(1/6)*hypergeom([5/6, 7/6],[11/6],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b*c)/(
d*x+c)^(1/6)

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Rubi [A]
time = 0.01, antiderivative size = 81, 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 (a+b x)^{5/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {11}{6};-\frac {d (a+b x)}{b c-a d}\right )}{5 \sqrt [6]{c+d x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(a + b*x)^(5/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[5/6, 7/6, 11/6, -((d*(a + b*x))/(b*c -
a*d))])/(5*(b*c - a*d)*(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 {1}{\sqrt [6]{a+b x} (c+d x)^{7/6}} \, dx &=\frac {\left (b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt [6]{a+b x} \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)^{5/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {11}{6};-\frac {d (a+b x)}{b c-a d}\right )}{5 (b c-a d) \sqrt [6]{c+d x}}\\ \end {align*}

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Mathematica [A]
time = 10.03, size = 73, normalized size = 0.90 \begin {gather*} \frac {6 (a+b x)^{5/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {11}{6};\frac {d (a+b x)}{-b c+a d}\right )}{5 b (c+d x)^{7/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(a + b*x)^(5/6)*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[5/6, 7/6, 11/6, (d*(a + b*x))/(-(b*c) +
a*d)])/(5*b*(c + d*x)^(7/6))

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {7}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [6]{a + b x} \left (c + d x\right )^{\frac {7}{6}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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