3.923 \(\int \frac{(a+b x)^n (c+d x)^3}{x^3} \, dx\)
Optimal. Leaf size=176 \[ -\frac{c (a+b x)^{n+1} \left (x \left (4 a^2 d^2+6 a b c d (n+1)-b^2 c^2 \left (1-n^2\right )\right )+a c (2 a d+b c (n+1))\right )}{2 a^2 b (n+1) x^2}-\frac{c (a+b x)^{n+1} \left (6 a^2 d^2+6 a b c d n-b^2 c^2 (1-n) n\right ) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{2 a^3 (n+1)}+\frac{d (c+d x)^2 (a+b x)^{n+1}}{b (n+1) x^2} \]
[Out]
(d*(a + b*x)^(1 + n)*(c + d*x)^2)/(b*(1 + n)*x^2) - (c*(a + b*x)^(1 + n)*(a*c*(2
*a*d + b*c*(1 + n)) + (4*a^2*d^2 + 6*a*b*c*d*(1 + n) - b^2*c^2*(1 - n^2))*x))/(2
*a^2*b*(1 + n)*x^2) - (c*(6*a^2*d^2 + 6*a*b*c*d*n - b^2*c^2*(1 - n)*n)*(a + b*x)
^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(2*a^3*(1 + n))
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Rubi [A] time = 0.476844, antiderivative size = 176, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167
\[ -\frac{c (a+b x)^{n+1} \left (x \left (4 a^2 d^2+6 a b c d (n+1)-b^2 c^2 \left (1-n^2\right )\right )+a c (2 a d+b c (n+1))\right )}{2 a^2 b (n+1) x^2}-\frac{c (a+b x)^{n+1} \left (6 a^2 d^2+6 a b c d n-b^2 c^2 (1-n) n\right ) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{2 a^3 (n+1)}+\frac{d (c+d x)^2 (a+b x)^{n+1}}{b (n+1) x^2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^n*(c + d*x)^3)/x^3,x]
[Out]
(d*(a + b*x)^(1 + n)*(c + d*x)^2)/(b*(1 + n)*x^2) - (c*(a + b*x)^(1 + n)*(a*c*(2
*a*d + b*c*(1 + n)) + (4*a^2*d^2 + 6*a*b*c*d*(1 + n) - b^2*c^2*(1 - n^2))*x))/(2
*a^2*b*(1 + n)*x^2) - (c*(6*a^2*d^2 + 6*a*b*c*d*n - b^2*c^2*(1 - n)*n)*(a + b*x)
^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(2*a^3*(1 + n))
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Rubi in Sympy [A] time = 31.5126, size = 175, normalized size = 0.99 \[ \frac{d \left (a + b x\right )^{n + 1} \left (c + d x\right )^{2}}{b x^{2} \left (n + 1\right )} - \frac{c \left (a + b x\right )^{n + 1} \left (a c \left (2 a d + b c \left (n + 1\right )\right ) + x \left (2 a d \left (2 a d + b c \left (n + 1\right ) + b c \left (n + 3\right )\right ) - b c \left (- n + 1\right ) \left (2 a d + b c \left (n + 1\right )\right )\right )\right )}{2 a^{2} b x^{2} \left (n + 1\right )} - \frac{c \left (a + b x\right )^{n + 1} \left (6 a^{2} d^{2} + 6 a b c d n + b^{2} c^{2} n^{2} - b^{2} c^{2} n\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{2 a^{3} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x+c)**3/x**3,x)
[Out]
d*(a + b*x)**(n + 1)*(c + d*x)**2/(b*x**2*(n + 1)) - c*(a + b*x)**(n + 1)*(a*c*(
2*a*d + b*c*(n + 1)) + x*(2*a*d*(2*a*d + b*c*(n + 1) + b*c*(n + 3)) - b*c*(-n +
1)*(2*a*d + b*c*(n + 1))))/(2*a**2*b*x**2*(n + 1)) - c*(a + b*x)**(n + 1)*(6*a**
2*d**2 + 6*a*b*c*d*n + b**2*c**2*n**2 - b**2*c**2*n)*hyper((1, n + 1), (n + 2,),
1 + b*x/a)/(2*a**3*(n + 1))
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Mathematica [A] time = 0.489021, size = 171, normalized size = 0.97 \[ (a+b x)^n \left (\frac{c^3 \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (2-n,-n;3-n;-\frac{a}{b x}\right )}{(n-2) x^2}+\frac{3 c^2 d \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (1-n,-n;2-n;-\frac{a}{b x}\right )}{(n-1) x}+d^2 \left (\frac{3 c \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}+\frac{a d+b d x}{b n+b}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^n*(c + d*x)^3)/x^3,x]
[Out]
(a + b*x)^n*((3*c^2*d*Hypergeometric2F1[1 - n, -n, 2 - n, -(a/(b*x))])/((-1 + n)
*(1 + a/(b*x))^n*x) + (c^3*Hypergeometric2F1[2 - n, -n, 3 - n, -(a/(b*x))])/((-2
+ n)*(1 + a/(b*x))^n*x^2) + d^2*((a*d + b*d*x)/(b + b*n) + (3*c*Hypergeometric2
F1[-n, -n, 1 - n, -(a/(b*x))])/(n*(1 + a/(b*x))^n)))
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Maple [F] time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{3}}{{x}^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x+c)^3/x^3,x)
[Out]
int((b*x+a)^n*(d*x+c)^3/x^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*(b*x + a)^n/x^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}{\left (b x + a\right )}^{n}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*(b*x + a)^n/x^3,x, algorithm="fricas")
[Out]
integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*(b*x + a)^n/x^3, x)
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Sympy [A] time = 21.4056, size = 1868, normalized size = 10.61 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**n*(d*x+c)**3/x**3,x)
[Out]
-a**3*b**2*b**n*c**3*n**3*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1
)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gam
ma(n + 2)) + a**3*b**2*b**n*c**3*n**2*(a/b + x)**n*gamma(n + 1)/(-2*a**5*gamma(n
+ 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a**3*
b**2*b**n*c**3*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**
5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)
) - a**3*b**2*b**n*c**3*n*(a/b + x)**n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a*
*4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - 2*a**3*b**2*b**n*
c**3*(a/b + x)**n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) +
2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**3*b**n*c**3*n**3*x*(a/b + x)**
n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*
gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a**2*b**3*b**n*c**3*n**2
*x*(a/b + x)**n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2
*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a**2*b**3*b**n*c**3*n*x*(a/b + x)**n*ler
chphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma
(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**3*b**n*c**3*n*x*(a/b
+ x)**n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b*
*2*(a/b + x)**2*gamma(n + 2)) - 2*a**2*b**3*b**n*c**3*x*(a/b + x)**n*gamma(n + 1
)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gam
ma(n + 2)) + 2*a*b**4*b**n*c**3*n**3*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/
a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a*
*3*b**2*(a/b + x)**2*gamma(n + 2)) - a*b**4*b**n*c**3*n**2*(a/b + x)**2*(a/b + x
)**n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*
(a/b + x)**2*gamma(n + 2)) - 2*a*b**4*b**n*c**3*n*(a/b + x)**2*(a/b + x)**n*lerc
hphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(
n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a*b**4*b**n*c**3*(a/b + x)**2*
(a/b + x)**n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a*
*3*b**2*(a/b + x)**2*gamma(n + 2)) - b**5*b**n*c**3*n**3*(a/b + x)**3*(a/b + x)*
*n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x
*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + b**5*b**n*c**3*n*(a/b +
x)**3*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n
+ 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + 3*b**n
*c**2*d*n**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x*gamma(n
+ 2)) + 3*b**n*c**2*d*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/
(x*gamma(n + 2)) - 3*b**n*c**2*d*n*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) -
3*b**n*c**2*d*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - 3*b**n*c*d**2*n*(a/b
+ x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - 3*b**n*c*d**2*
(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + d**3*Piec
ewise((a**n*x, Eq(b, 0)), (Piecewise(((a + b*x)**(n + 1)/(n + 1), Ne(n, -1)), (l
og(a + b*x), True))/b, True)) + 3*b*b**n*c**2*d*n**2*(a/b + x)**n*lerchphi(1 + b
*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) + 3*b*b**n*c**2*d*n*(a/b + x)**n*l
erchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - 3*b*b**n*c**2*d*n*(
a/b + x)**n*gamma(n + 1)/(a*gamma(n + 2)) - 3*b*b**n*c**2*d*(a/b + x)**n*gamma(n
+ 1)/(a*gamma(n + 2)) - 3*b*b**n*c*d**2*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1,
n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - 3*b*b**n*c*d**2*x*(a/b + x)**n*lerchphi(
1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - 3*b**2*b**n*c**2*d*n**2*(a/
b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a**2*x*gamma(
n + 2)) - 3*b**2*b**n*c**2*d*n*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1,
n + 1)*gamma(n + 1)/(a**2*x*gamma(n + 2))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{3}{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*(b*x + a)^n/x^3,x, algorithm="giac")
[Out]
integrate((d*x + c)^3*(b*x + a)^n/x^3, x)