3.1792 \(\int (a+b x)^{7/6} (c+d x)^{13/6} \, dx\)

Optimal. Leaf size=84 \[ \frac {6 (a+b x)^{13/6} \sqrt [6]{c+d x} (b c-a d)^2 \, _2F_1\left (-\frac {13}{6},\frac {13}{6};\frac {19}{6};-\frac {d (a+b x)}{b c-a d}\right )}{13 b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]

[Out]

6/13*(-a*d+b*c)^2*(b*x+a)^(13/6)*(d*x+c)^(1/6)*hypergeom([-13/6, 13/6],[19/6],-d*(b*x+a)/(-a*d+b*c))/b^3/(b*(d
*x+c)/(-a*d+b*c))^(1/6)

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Rubi [A]  time = 0.02, 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 = {70, 69} \[ \frac {6 (a+b x)^{13/6} \sqrt [6]{c+d x} (b c-a d)^2 \, _2F_1\left (-\frac {13}{6},\frac {13}{6};\frac {19}{6};-\frac {d (a+b x)}{b c-a d}\right )}{13 b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(b*c - a*d)^2*(a + b*x)^(13/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-13/6, 13/6, 19/6, -((d*(a + b*x))/(b*c -
a*d))])/(13*b^3*((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 (a+b x)^{7/6} (c+d x)^{13/6} \, dx &=\frac {\left ((b c-a d)^2 \sqrt [6]{c+d x}\right ) \int (a+b x)^{7/6} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{13/6} \, dx}{b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {6 (b c-a d)^2 (a+b x)^{13/6} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {13}{6},\frac {13}{6};\frac {19}{6};-\frac {d (a+b x)}{b c-a d}\right )}{13 b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 73, normalized size = 0.87 \[ \frac {6 (a+b x)^{13/6} (c+d x)^{13/6} \, _2F_1\left (-\frac {13}{6},\frac {13}{6};\frac {19}{6};\frac {d (a+b x)}{a d-b c}\right )}{13 b \left (\frac {b (c+d x)}{b c-a d}\right )^{13/6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(a + b*x)^(13/6)*(c + d*x)^(13/6)*Hypergeometric2F1[-13/6, 13/6, 19/6, (d*(a + b*x))/(-(b*c) + a*d)])/(13*b
*((b*(c + d*x))/(b*c - a*d))^(13/6))

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fricas [F]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b d^{2} x^{3} + a c^{2} + {\left (2 \, b c d + a d^{2}\right )} x^{2} + {\left (b c^{2} + 2 \, a c d\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {1}{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)*(d*x+c)^(13/6),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {13}{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)*(d*x+c)^(13/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(7/6)*(d*x + c)^(13/6), x)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{\frac {7}{6}} \left (d x +c \right )^{\frac {13}{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(7/6)*(d*x+c)^(13/6),x)

[Out]

int((b*x+a)^(7/6)*(d*x+c)^(13/6),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {13}{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)*(d*x+c)^(13/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/6)*(d*x + c)^(13/6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,x\right )}^{7/6}\,{\left (c+d\,x\right )}^{13/6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(7/6)*(c + d*x)^(13/6),x)

[Out]

int((a + b*x)^(7/6)*(c + d*x)^(13/6), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(7/6)*(d*x+c)**(13/6),x)

[Out]

Timed out

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