3.1883 \(\int (a+b x)^{-1-\frac{b c}{b c-a d}} (c+d x)^{-1+\frac{a d}{b c-a d}} \, dx\)

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 = 0.0790921, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{(a+b x)^{-\frac{a d}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{a b c}-\frac{(a+b x)^{-\frac{b c}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{b c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(-1 - (b*c)/(b*c - a*d))*(c + d*x)^(-1 + (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 = 31.1825, size = 104, normalized size = 1.07 \[ - \frac{\left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{b c}{- a d + b c}} \left (a + b x\right )^{\frac{b c}{a d - b c}} \left (c + d x\right )^{\frac{a d}{- a d + b c}}{{}_{2}F_{1}\left (\begin{matrix} \frac{b c}{- a d + b c} + 1, \frac{a d}{- a d + b c} \\ - \frac{b c}{a d - b c} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(-1-b*c/(-a*d+b*c))*(d*x+c)**(-1+a*d/(-a*d+b*c)),x)

[Out]

-(d*(a + b*x)/(a*d - b*c))**(b*c/(-a*d + b*c))*(a + b*x)**(b*c/(a*d - b*c))*(c +
d*x)**(a*d/(-a*d + b*c))*hyper((b*c/(-a*d + b*c) + 1, a*d/(-a*d + b*c)), (-b*c/
(a*d - b*c),), b*(-c - d*x)/(a*d - b*c))/(a*d)

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Mathematica [A]  time = 0.330294, size = 46, normalized size = 0.47 \[ \frac{x (a+b x)^{\frac{b c}{a d-b c}} (c+d x)^{\frac{a d}{b c-a d}}}{a c} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(-1 - (b*c)/(b*c - a*d))*(c + d*x)^(-1 + (a*d)/(b*c - a*d)),x]

[Out]

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

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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)^(-1-b*c/(-a*d+b*c))*(d*x+c)^(-1+a*d/(-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. \[ \int{\left (b x + a\right )}^{-\frac{b c}{b c - a d} - 1}{\left (d x + c\right )}^{\frac{a d}{b c - a d} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233403, 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((b*x + a)^(-b*c/(b*c - a*d) - 1)*(d*x + c)^(a*d/(b*c - a*d) - 1),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)**(-1-b*c/(-a*d+b*c))*(d*x+c)**(-1+a*d/(-a*d+b*c)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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