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

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

[Out]

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

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Rubi [A]  time = 0.11613, 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 \[ -\frac{(a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac{d (a+b x)}{b c-a d},\frac{a+b x}{a}\right )}{a (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.5235, size = 63, normalized size = 0.74 \[ - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- p} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{p} \operatorname{appellf_{1}}{\left (n + 1,1,- p,n + 2,\frac{a + b x}{a},\frac{d \left (a + b x\right )}{a d - b c} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.38233, size = 214, normalized size = 2.52 \[ \frac{b d x (n+p-1) (a+b x)^n (c+d x)^p F_1\left (-n-p;-n,-p;-n-p+1;-\frac{a}{b x},-\frac{c}{d x}\right )}{(n+p) \left (b d x (n+p-1) F_1\left (-n-p;-n,-p;-n-p+1;-\frac{a}{b x},-\frac{c}{d x}\right )-a d n F_1\left (-n-p+1;1-n,-p;-n-p+2;-\frac{a}{b x},-\frac{c}{d x}\right )-b c p F_1\left (-n-p+1;-n,1-p;-n-p+2;-\frac{a}{b x},-\frac{c}{d x}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)^p/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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