3.1884 \(\int (a+b x)^{\frac{-2 b c+a d}{b c-a d}} (c+d x)^{\frac{b c-2 a d}{-b c+a d}} \, dx\)

Optimal. Leaf size=97 \[ \frac{(a+b x)^{-\frac{a d}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{a b c}-\frac{(a+b x)^{-\frac{b c}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{b c} \]

[Out]

-((c + d*x)^((a*d)/(b*c - a*d))/(b*c*(a + b*x)^((b*c)/(b*c - a*d)))) + (c + d*x)
^((a*d)/(b*c - a*d))/(a*b*c*(a + b*x)^((a*d)/(b*c - a*d)))

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Rubi [A]  time = 0.0792646, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.039 \[ \frac{(a+b x)^{-\frac{a d}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{a b c}-\frac{(a+b x)^{-\frac{b c}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{b c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^((-2*b*c + a*d)/(b*c - a*d))*(c + d*x)^((b*c - 2*a*d)/(-(b*c) + a*d)),x]

[Out]

-((c + d*x)^((a*d)/(b*c - a*d))/(b*c*(a + b*x)^((b*c)/(b*c - a*d)))) + (c + d*x)
^((a*d)/(b*c - a*d))/(a*b*c*(a + b*x)^((a*d)/(b*c - a*d)))

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Rubi in Sympy [A]  time = 33.0808, size = 128, normalized size = 1.32 \[ - \frac{\left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{a d - 2 b c}{a d - b c}} \left (a + b x\right )^{\frac{a d - 2 b c}{- a d + b c}} \left (c + d x\right )^{- \frac{a d}{a d - b c}} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{a d - 2 b c}{a d - b c}, - \frac{a d}{a d - b c} \\ - \frac{b c}{a d - b c} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{a d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)**((-2*a*d+b*c)/(a*d-b*c)),x)

[Out]

-(d*(a + b*x)/(a*d - b*c))**((a*d - 2*b*c)/(a*d - b*c))*(a + b*x)**((a*d - 2*b*c
)/(-a*d + b*c))*(c + d*x)**(-a*d/(a*d - b*c))*(a*d - b*c)*hyper(((a*d - 2*b*c)/(
a*d - b*c), -a*d/(a*d - b*c)), (-b*c/(a*d - b*c),), b*(-c - d*x)/(a*d - b*c))/(a
*d**2)

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Mathematica [C]  time = 0.218624, size = 159, normalized size = 1.64 \[ \frac{(b c-a d) (a+b x)^{\frac{a d-2 b c}{b c-a d}} \left (\frac{d (a+b x)}{a d-b c}\right )^{\frac{a d-2 b c}{a d-b c}} (c+d x)^{\frac{a d}{b c-a d}} \, _2F_1\left (\frac{a d}{b c-a d},\frac{a d-2 b c}{a d-b c};\frac{b c}{b c-a d};\frac{b (c+d x)}{b c-a d}\right )}{a d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^((-2*b*c + a*d)/(b*c - a*d))*(c + d*x)^((b*c - 2*a*d)/(-(b*c) + a*d)),x]

[Out]

((b*c - a*d)*(a + b*x)^((-2*b*c + a*d)/(b*c - a*d))*((d*(a + b*x))/(-(b*c) + a*d
))^((-2*b*c + a*d)/(-(b*c) + a*d))*(c + d*x)^((a*d)/(b*c - a*d))*Hypergeometric2
F1[(a*d)/(b*c - a*d), (-2*b*c + a*d)/(-(b*c) + a*d), (b*c)/(b*c - a*d), (b*(c +
d*x))/(b*c - a*d)])/(a*d^2)

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Maple [A]  time = 0.007, size = 66, normalized size = 0.7 \[{\frac{x}{ac} \left ( bx+a \right ) ^{1-{\frac{ad-2\,bc}{ad-bc}}} \left ( dx+c \right ) ^{1-{\frac{2\,ad-bc}{ad-bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)^((-2*a*d+b*c)/(a*d-b*c)),x)

[Out]

(b*x+a)^(1-(a*d-2*b*c)/(a*d-b*c))*(d*x+c)^(1-(2*a*d-b*c)/(a*d-b*c))/a/c*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{-\frac{2 \, b c - a d}{b c - a d}}{\left (d x + c\right )}^{-\frac{b c - 2 \, a d}{b c - a d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-(2*b*c - a*d)/(b*c - a*d))*(d*x + c)^(-(b*c - 2*a*d)/(b*c - a*d)),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(-(2*b*c - a*d)/(b*c - a*d))*(d*x + c)^(-(b*c - 2*a*d)/(b*c
- a*d)), x)

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Fricas [A]  time = 0.233228, size = 113, normalized size = 1.16 \[ \frac{b d x^{3} + a c x +{\left (b c + a d\right )} x^{2}}{{\left (b x + a\right )}^{\frac{2 \, b c - a d}{b c - a d}}{\left (d x + c\right )}^{\frac{b c - 2 \, a d}{b c - a d}} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^((2*b*c - a*d)/(b*c - a*d))*(d*x + c)^((b*c - 2*a*d)/(b*c - a*d))),x, algorithm="fricas")

[Out]

(b*d*x^3 + a*c*x + (b*c + a*d)*x^2)/((b*x + a)^((2*b*c - a*d)/(b*c - a*d))*(d*x
+ c)^((b*c - 2*a*d)/(b*c - a*d))*a*c)

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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)**((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)**((-2*a*d+b*c)/(a*d-b*c)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{2 \, b c - a d}{b c - a d}}{\left (d x + c\right )}^{\frac{b c - 2 \, a d}{b c - a d}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^((2*b*c - a*d)/(b*c - a*d))*(d*x + c)^((b*c - 2*a*d)/(b*c - a*d))),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^((2*b*c - a*d)/(b*c - a*d))*(d*x + c)^((b*c - 2*a*d)/(b*c
- a*d))), x)