3.983 \(\int x^m (a+b x)^{1+n} (c+d x)^n \, dx\)

Optimal. Leaf size=79 \[ \frac{a x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n-1,-n;m+2;-\frac{b x}{a},-\frac{d x}{c}\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.149882, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n-1,-n;m+2;-\frac{b x}{a},-\frac{d x}{c}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x)^(1 + n)*(c + d*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.9235, size = 61, normalized size = 0.77 \[ \frac{a x^{m + 1} \left (1 + \frac{b x}{a}\right )^{- n} \left (1 + \frac{d x}{c}\right )^{- n} \left (a + b x\right )^{n} \left (c + d x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,- n,- n - 1,m + 2,- \frac{d x}{c},- \frac{b x}{a} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x+a)**(1+n)*(d*x+c)**n,x)

[Out]

a*x**(m + 1)*(1 + b*x/a)**(-n)*(1 + d*x/c)**(-n)*(a + b*x)**n*(c + d*x)**n*appel
lf1(m + 1, -n, -n - 1, m + 2, -d*x/c, -b*x/a)/(m + 1)

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Mathematica [B]  time = 1.07948, size = 308, normalized size = 3.9 \[ \frac{a c x^{m+1} (a+b x)^n (c+d x)^n \left (\frac{a (m+2)^2 F_1\left (m+1;-n,-n;m+2;-\frac{b x}{a},-\frac{d x}{c}\right )}{(m+1) \left (a c (m+2) F_1\left (m+1;-n,-n;m+2;-\frac{b x}{a},-\frac{d x}{c}\right )+n x \left (b c F_1\left (m+2;1-n,-n;m+3;-\frac{b x}{a},-\frac{d x}{c}\right )+a d F_1\left (m+2;-n,1-n;m+3;-\frac{b x}{a},-\frac{d x}{c}\right )\right )\right )}+\frac{b (m+3) x F_1\left (m+2;-n,-n;m+3;-\frac{b x}{a},-\frac{d x}{c}\right )}{a c (m+3) F_1\left (m+2;-n,-n;m+3;-\frac{b x}{a},-\frac{d x}{c}\right )+n x \left (b c F_1\left (m+3;1-n,-n;m+4;-\frac{b x}{a},-\frac{d x}{c}\right )+a d F_1\left (m+3;-n,1-n;m+4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}\right )}{m+2} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*c*x^(1 + m)*(a + b*x)^n*(c + d*x)^n*((a*(2 + m)^2*AppellF1[1 + m, -n, -n, 2 +
m, -((b*x)/a), -((d*x)/c)])/((1 + m)*(a*c*(2 + m)*AppellF1[1 + m, -n, -n, 2 + m
, -((b*x)/a), -((d*x)/c)] + n*x*(b*c*AppellF1[2 + m, 1 - n, -n, 3 + m, -((b*x)/a
), -((d*x)/c)] + a*d*AppellF1[2 + m, -n, 1 - n, 3 + m, -((b*x)/a), -((d*x)/c)]))
) + (b*(3 + m)*x*AppellF1[2 + m, -n, -n, 3 + m, -((b*x)/a), -((d*x)/c)])/(a*c*(3
+ m)*AppellF1[2 + m, -n, -n, 3 + m, -((b*x)/a), -((d*x)/c)] + n*x*(b*c*AppellF1
[3 + m, 1 - n, -n, 4 + m, -((b*x)/a), -((d*x)/c)] + a*d*AppellF1[3 + m, -n, 1 -
n, 4 + m, -((b*x)/a), -((d*x)/c)]))))/(2 + m)

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Maple [F]  time = 0.214, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( bx+a \right ) ^{1+n} \left ( dx+c \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x)

[Out]

int(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n + 1}{\left (d x + c\right )}^{n} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^(n + 1)*(d*x + c)^n*x^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^(n + 1)*(d*x + c)^n*x^m, 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(x**m*(b*x+a)**(1+n)*(d*x+c)**n,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^(n + 1)*(d*x + c)^n*x^m, x)