3.1819 \(\int \frac {(c+d x)^{7/6}}{(a+b x)^{5/6}} \, dx\)

Optimal. Leaf size=80 \[ \frac {6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac {7}{6},\frac {1}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]

[Out]

6*(-a*d+b*c)*(b*x+a)^(1/6)*(d*x+c)^(1/6)*hypergeom([-7/6, 1/6],[7/6],-d*(b*x+a)/(-a*d+b*c))/b^2/(b*(d*x+c)/(-a
*d+b*c))^(1/6)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 80, 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} \[ \frac {6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac {7}{6},\frac {1}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-7/6, 1/6, 7/6, -((d*(a + b*x))/(b*c - a*d))]
)/(b^2*((b*(c + d*x))/(b*c - a*d))^(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 {(c+d x)^{7/6}}{(a+b x)^{5/6}} \, dx &=\frac {\left ((b c-a d) \sqrt [6]{c+d x}\right ) \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{7/6}}{(a+b x)^{5/6}} \, dx}{b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {6 (b c-a d) \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {7}{6},\frac {1}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 71, normalized size = 0.89 \[ \frac {6 \sqrt [6]{a+b x} (c+d x)^{7/6} \, _2F_1\left (-\frac {7}{6},\frac {1}{6};\frac {7}{6};\frac {d (a+b x)}{a d-b c}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )}^{\frac {7}{6}}}{{\left (b x + a\right )}^{\frac {5}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{\frac {7}{6}}}{\left (b x +a \right )^{\frac {5}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________