3.921 \(\int \frac{(a+b x)^n (c+d x)^3}{x} \, dx\)

Optimal. Leaf size=131 \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) (a+b x)^{n+1}}{b^3 (n+1)}+\frac{d^2 (3 b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d^3 (a+b x)^{n+3}}{b^3 (n+3)}-\frac{c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

(d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*(a + b*x)^(1 + n))/(b^3*(1 + n)) + (d^2*(3*
b*c - 2*a*d)*(a + b*x)^(2 + n))/(b^3*(2 + n)) + (d^3*(a + b*x)^(3 + n))/(b^3*(3
+ n)) - (c^3*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/
(a*(1 + n))

_______________________________________________________________________________________

Rubi [A]  time = 0.14899, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) (a+b x)^{n+1}}{b^3 (n+1)}+\frac{d^2 (3 b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d^3 (a+b x)^{n+3}}{b^3 (n+3)}-\frac{c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^n*(c + d*x)^3)/x,x]

[Out]

(d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*(a + b*x)^(1 + n))/(b^3*(1 + n)) + (d^2*(3*
b*c - 2*a*d)*(a + b*x)^(2 + n))/(b^3*(2 + n)) + (d^3*(a + b*x)^(3 + n))/(b^3*(3
+ n)) - (c^3*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/
(a*(1 + n))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.0793, size = 116, normalized size = 0.89 \[ \frac{d^{3} \left (a + b x\right )^{n + 3}}{b^{3} \left (n + 3\right )} - \frac{d^{2} \left (a + b x\right )^{n + 2} \left (2 a d - 3 b c\right )}{b^{3} \left (n + 2\right )} + \frac{d \left (a + b x\right )^{n + 1} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{b^{3} \left (n + 1\right )} - \frac{c^{3} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n*(d*x+c)**3/x,x)

[Out]

d**3*(a + b*x)**(n + 3)/(b**3*(n + 3)) - d**2*(a + b*x)**(n + 2)*(2*a*d - 3*b*c)
/(b**3*(n + 2)) + d*(a + b*x)**(n + 1)*(a**2*d**2 - 3*a*b*c*d + 3*b**2*c**2)/(b*
*3*(n + 1)) - c**3*(a + b*x)**(n + 1)*hyper((1, n + 1), (n + 2,), 1 + b*x/a)/(a*
(n + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.614527, size = 251, normalized size = 1.92 \[ (a+b x)^n \left (\frac{d \left (\frac{b x}{a}+1\right )^{-n} \left (2 a^3 d^2 \left (\left (\frac{b x}{a}+1\right )^n-1\right )-a^2 b d \left (3 c (n+3) \left (\left (\frac{b x}{a}+1\right )^n-1\right )+2 d n x \left (\frac{b x}{a}+1\right )^n\right )+b^3 x \left (\frac{b x}{a}+1\right )^n \left (3 c^2 \left (n^2+5 n+6\right )+3 c d \left (n^2+4 n+3\right ) x+d^2 \left (n^2+3 n+2\right ) x^2\right )+a b^2 \left (\frac{b x}{a}+1\right )^n \left (3 c^2 \left (n^2+5 n+6\right )+3 c d n (n+3) x+d^2 n (n+1) x^2\right )\right )}{b^3 (n+1) (n+2) (n+3)}+\frac{c^3 \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^n*(c + d*x)^3)/x,x]

[Out]

(a + b*x)^n*((d*(a*b^2*(1 + (b*x)/a)^n*(3*c^2*(6 + 5*n + n^2) + 3*c*d*n*(3 + n)*
x + d^2*n*(1 + n)*x^2) + b^3*x*(1 + (b*x)/a)^n*(3*c^2*(6 + 5*n + n^2) + 3*c*d*(3
+ 4*n + n^2)*x + d^2*(2 + 3*n + n^2)*x^2) + 2*a^3*d^2*(-1 + (1 + (b*x)/a)^n) -
a^2*b*d*(2*d*n*x*(1 + (b*x)/a)^n + 3*c*(3 + n)*(-1 + (1 + (b*x)/a)^n))))/(b^3*(1
+ n)*(2 + n)*(3 + n)*(1 + (b*x)/a)^n) + (c^3*Hypergeometric2F1[-n, -n, 1 - n, -
(a/(b*x))])/(n*(1 + a/(b*x))^n))

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{3}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)^3/x,x)

[Out]

int((b*x+a)^n*(d*x+c)^3/x,x)

_______________________________________________________________________________________

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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}{\left (b x + a\right )}^{n}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3*(b*x + a)^n/x,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, x)

_______________________________________________________________________________________

Sympy [A]  time = 14.423, size = 993, normalized size = 7.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n*(d*x+c)**3/x,x)

[Out]

-b**n*c**3*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2
) - b**n*c**3*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n +
2) + 3*c**2*d*Piecewise((a**n*x, Eq(b, 0)), (Piecewise(((a + b*x)**(n + 1)/(n +
1), Ne(n, -1)), (log(a + b*x), True))/b, True)) + 3*c*d**2*Piecewise((a**n*x**2/
2, Eq(b, 0)), (a*log(a/b + x)/(a*b**2 + b**3*x) + a/(a*b**2 + b**3*x) + b*x*log(
a/b + x)/(a*b**2 + b**3*x), Eq(n, -2)), (-a*log(a/b + x)/b**2 + x/b, Eq(n, -1)),
(-a**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + a*b*n*x*(a + b*x)**n/(b**
2*n**2 + 3*b**2*n + 2*b**2) + b**2*n*x**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2
*b**2) + b**2*x**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2), True)) + d**3*P
iecewise((a**n*x**3/3, Eq(b, 0)), (2*a**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x
+ 2*b**5*x**2) + a**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*x*log(a/
b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 2*b**2*x**2*log(a/b + x)/(2*a*
*2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - 2*b**2*x**2/(2*a**2*b**3 + 4*a*b**4*x + 2*
b**5*x**2), Eq(n, -3)), (-2*a**2*log(a/b + x)/(a*b**3 + b**4*x) - 2*a**2/(a*b**3
+ b**4*x) - 2*a*b*x*log(a/b + x)/(a*b**3 + b**4*x) + b**2*x**2/(a*b**3 + b**4*x
), Eq(n, -2)), (a**2*log(a/b + x)/b**3 - a*x/b**2 + x**2/(2*b), Eq(n, -1)), (2*a
**3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 2*a**2*b*n*x*(
a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*n**2*x**2*(a
+ b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*n*x**2*(a + b
*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + b**3*n**2*x**3*(a + b*x)
**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 3*b**3*n*x**3*(a + b*x)**n/
(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 2*b**3*x**3*(a + b*x)**n/(b**3*
n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3), True)) - b*b**n*c**3*n*x*(a/b + x)**n*
lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b*b**n*c**3*x*(a/b
+ x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{3}{\left (b x + a\right )}^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3*(b*x + a)^n/x,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*(b*x + a)^n/x, x)