3.19.18 \(\int \frac {(c+d x)^{13/6}}{(a+b x)^{5/6}} \, dx\) [1818]

Optimal. Leaf size=82 \[ \frac {6 (b c-a d)^2 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {13}{6},\frac {1}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]

[Out]

6*(-a*d+b*c)^2*(b*x+a)^(1/6)*(d*x+c)^(1/6)*hypergeom([-13/6, 1/6],[7/6],-d*(b*x+a)/(-a*d+b*c))/b^3/(b*(d*x+c)/
(-a*d+b*c))^(1/6)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 82, 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 \sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^2 \, _2F_1\left (-\frac {13}{6},\frac {1}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(13/6)/(a + b*x)^(5/6),x]

[Out]

(6*(b*c - a*d)^2*(a + b*x)^(1/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-13/6, 1/6, 7/6, -((d*(a + b*x))/(b*c - a*d
))])/(b^3*((b*(c + d*x))/(b*c - a*d))^(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 {(c+d x)^{13/6}}{(a+b x)^{5/6}} \, dx &=\frac {\left ((b c-a d)^2 \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 )^{13/6}}{(a+b x)^{5/6}} \, dx}{b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {6 (b c-a d)^2 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {13}{6},\frac {1}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 10.06, size = 71, normalized size = 0.87 \begin {gather*} \frac {6 \sqrt [6]{a+b x} (c+d x)^{13/6} \, _2F_1\left (-\frac {13}{6},\frac {1}{6};\frac {7}{6};\frac {d (a+b x)}{-b c+a d}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{13/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(13/6)/(a + b*x)^(5/6),x]

[Out]

(6*(a + b*x)^(1/6)*(c + d*x)^(13/6)*Hypergeometric2F1[-13/6, 1/6, 7/6, (d*(a + b*x))/(-(b*c) + a*d)])/(b*((b*(
c + d*x))/(b*c - a*d))^(13/6))

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {13}{6}}}{\left (b x +a \right )^{\frac {5}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(13/6)/(b*x+a)^(5/6),x)

[Out]

int((d*x+c)^(13/6)/(b*x+a)^(5/6),x)

________________________________________________________________________________________

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)^(13/6)/(b*x+a)^(5/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(13/6)/(b*x + a)^(5/6), x)

________________________________________________________________________________________

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)^(13/6)/(b*x+a)^(5/6),x, algorithm="fricas")

[Out]

integral((d^2*x^2 + 2*c*d*x + c^2)*(d*x + c)^(1/6)/(b*x + a)^(5/6), x)

________________________________________________________________________________________

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((d*x+c)**(13/6)/(b*x+a)**(5/6),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5457 deep

________________________________________________________________________________________

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)^(13/6)/(b*x+a)^(5/6),x, algorithm="giac")

[Out]

integrate((d*x + c)^(13/6)/(b*x + a)^(5/6), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{13/6}}{{\left (a+b\,x\right )}^{5/6}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(13/6)/(a + b*x)^(5/6),x)

[Out]

int((c + d*x)^(13/6)/(a + b*x)^(5/6), x)

________________________________________________________________________________________