3.950 \(\int \frac{(a+b x)^n (c+d x)^p}{x^2} \, dx\)

Optimal. Leaf size=85 \[ \frac{b (a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac{d (a+b x)}{b c-a d},\frac{a+b x}{a}\right )}{a^2 (n+1)} \]

[Out]

(b*(a + b*x)^(1 + n)*(c + d*x)^p*AppellF1[1 + n, -p, 2, 2 + n, -((d*(a + b*x))/(
b*c - a*d)), (a + b*x)/a])/(a^2*(1 + n)*((b*(c + d*x))/(b*c - a*d))^p)

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Rubi [A]  time = 0.121609, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{b (a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac{d (a+b x)}{b c-a d},\frac{a+b x}{a}\right )}{a^2 (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(b*(a + b*x)^(1 + n)*(c + d*x)^p*AppellF1[1 + n, -p, 2, 2 + n, -((d*(a + b*x))/(
b*c - a*d)), (a + b*x)/a])/(a^2*(1 + n)*((b*(c + d*x))/(b*c - a*d))^p)

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Rubi in Sympy [A]  time = 17.4961, size = 65, normalized size = 0.76 \[ \frac{b \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- p} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{p} \operatorname{appellf_{1}}{\left (n + 1,2,- p,n + 2,\frac{a + b x}{a},\frac{d \left (a + b x\right )}{a d - b c} \right )}}{a^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*(b*(-c - d*x)/(a*d - b*c))**(-p)*(a + b*x)**(n + 1)*(c + d*x)**p*appellf1(n +
1, 2, -p, n + 2, (a + b*x)/a, d*(a + b*x)/(a*d - b*c))/(a**2*(n + 1))

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Mathematica [B]  time = 0.427185, size = 216, normalized size = 2.54 \[ \frac{b d (n+p-2) (a+b x)^n (c+d x)^p F_1\left (-n-p+1;-n,-p;-n-p+2;-\frac{a}{b x},-\frac{c}{d x}\right )}{(n+p-1) \left (b d x (n+p-2) F_1\left (-n-p+1;-n,-p;-n-p+2;-\frac{a}{b x},-\frac{c}{d x}\right )-a d n F_1\left (-n-p+2;1-n,-p;-n-p+3;-\frac{a}{b x},-\frac{c}{d x}\right )-b c p F_1\left (-n-p+2;-n,1-p;-n-p+3;-\frac{a}{b x},-\frac{c}{d x}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(b*d*(-2 + n + p)*(a + b*x)^n*(c + d*x)^p*AppellF1[1 - n - p, -n, -p, 2 - n - p,
-(a/(b*x)), -(c/(d*x))])/((-1 + n + p)*(b*d*(-2 + n + p)*x*AppellF1[1 - n - p,
-n, -p, 2 - n - p, -(a/(b*x)), -(c/(d*x))] - a*d*n*AppellF1[2 - n - p, 1 - n, -p
, 3 - n - p, -(a/(b*x)), -(c/(d*x))] - b*c*p*AppellF1[2 - n - p, -n, 1 - p, 3 -
n - p, -(a/(b*x)), -(c/(d*x))]))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)^p/x^2,x)

[Out]

int((b*x+a)^n*(d*x+c)^p/x^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n*(d*x + c)^p/x^2, 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)**n*(d*x+c)**p/x**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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