3.1793 \(\int (a+b x)^{7/6} (c+d x)^{13/6} \, dx\)

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*(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))

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Rubi [A]  time = 0.0907859, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{6 (a+b x)^{13/6} \sqrt [6]{c+d x} (b c-a d)^2 \, _2F_1\left (-\frac{13}{6},\frac{13}{6};\frac{19}{6};-\frac{d (a+b x)}{b c-a d}\right )}{13 b^3 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]

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))

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Rubi in Sympy [A]  time = 13.6823, size = 70, normalized size = 0.83 \[ \frac{6 \sqrt [6]{a + b x} \left (c + d x\right )^{\frac{19}{6}} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{6}, \frac{19}{6} \\ \frac{25}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{19 d^{2} \sqrt [6]{\frac{d \left (a + b x\right )}{a d - b c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(a + b*x)**(1/6)*(c + d*x)**(19/6)*(a*d - b*c)*hyper((-7/6, 19/6), (25/6,), b*
(-c - d*x)/(a*d - b*c))/(19*d**2*(d*(a + b*x)/(a*d - b*c))**(1/6))

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Mathematica [B]  time = 0.459845, size = 234, normalized size = 2.79 \[ -\frac{3 \sqrt [6]{c+d x} \left (-d (a+b x) \left (91 a^4 d^4-26 a^3 b d^3 (15 c+d x)+2 a^2 b^2 d^2 \left (320 c^2+55 c d x+8 d^2 x^2\right )+2 a b^3 d \left (195 c^3+1225 c^2 d x+1280 c d^2 x^2+432 d^3 x^3\right )+b^4 \left (-91 c^4+26 c^3 d x+1264 c^2 d^2 x^2+1696 c d^3 x^3+640 d^4 x^4\right )\right )-91 (b c-a d)^5 \left (\frac{d (a+b x)}{a d-b c}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{8320 b^3 d^3 (a+b x)^{5/6}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(c + d*x)^(1/6)*(-(d*(a + b*x)*(91*a^4*d^4 - 26*a^3*b*d^3*(15*c + d*x) + 2*a
^2*b^2*d^2*(320*c^2 + 55*c*d*x + 8*d^2*x^2) + 2*a*b^3*d*(195*c^3 + 1225*c^2*d*x
+ 1280*c*d^2*x^2 + 432*d^3*x^3) + b^4*(-91*c^4 + 26*c^3*d*x + 1264*c^2*d^2*x^2 +
1696*c*d^3*x^3 + 640*d^4*x^4))) - 91*(b*c - a*d)^5*((d*(a + b*x))/(-(b*c) + a*d
))^(5/6)*Hypergeometric2F1[1/6, 5/6, 7/6, (b*(c + d*x))/(b*c - a*d)]))/(8320*b^3
*d^3*(a + b*x)^(5/6))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{7}{6}}} \left ( dx+c \right ) ^{{\frac{13}{6}}}\, dx \]

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

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

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

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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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out