3.358 \(\int \frac{(a+b x)^n \left (c+d x^2\right )^2}{x} \, dx\)
Optimal. Leaf size=148 \[ -\frac{a d \left (a^2 d+2 b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{d \left (3 a^2 d+2 b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d^2 (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d^2 (a+b x)^{n+4}}{b^4 (n+4)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]
[Out]
-((a*d*(2*b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^4*(1 + n))) + (d*(2*b^2*c + 3*a^2
*d)*(a + b*x)^(2 + n))/(b^4*(2 + n)) - (3*a*d^2*(a + b*x)^(3 + n))/(b^4*(3 + n))
+ (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n)) - (c^2*(a + b*x)^(1 + n)*Hypergeometric
2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*(1 + n))
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Rubi [A] time = 0.179472, antiderivative size = 148, 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
\[ -\frac{a d \left (a^2 d+2 b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{d \left (3 a^2 d+2 b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d^2 (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d^2 (a+b x)^{n+4}}{b^4 (n+4)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^n*(c + d*x^2)^2)/x,x]
[Out]
-((a*d*(2*b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^4*(1 + n))) + (d*(2*b^2*c + 3*a^2
*d)*(a + b*x)^(2 + n))/(b^4*(2 + n)) - (3*a*d^2*(a + b*x)^(3 + n))/(b^4*(3 + n))
+ (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n)) - (c^2*(a + b*x)^(1 + n)*Hypergeometric
2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*(1 + n))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 32.3735, size = 129, normalized size = 0.87 \[ - \frac{3 a d^{2} \left (a + b x\right )^{n + 3}}{b^{4} \left (n + 3\right )} - \frac{a d \left (a + b x\right )^{n + 1} \left (a^{2} d + 2 b^{2} c\right )}{b^{4} \left (n + 1\right )} + \frac{d^{2} \left (a + b x\right )^{n + 4}}{b^{4} \left (n + 4\right )} + \frac{d \left (a + b x\right )^{n + 2} \left (3 a^{2} d + 2 b^{2} c\right )}{b^{4} \left (n + 2\right )} - \frac{c^{2} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x**2+c)**2/x,x)
[Out]
-3*a*d**2*(a + b*x)**(n + 3)/(b**4*(n + 3)) - a*d*(a + b*x)**(n + 1)*(a**2*d + 2
*b**2*c)/(b**4*(n + 1)) + d**2*(a + b*x)**(n + 4)/(b**4*(n + 4)) + d*(a + b*x)**
(n + 2)*(3*a**2*d + 2*b**2*c)/(b**4*(n + 2)) - c**2*(a + b*x)**(n + 1)*hyper((1,
n + 1), (n + 2,), 1 + b*x/a)/(a*(n + 1))
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Mathematica [A] time = 0.539422, size = 259, normalized size = 1.75 \[ (a+b x)^n \left (\frac{2 c d \left (a^2 \left (\left (\frac{b x}{a}+1\right )^{-n}-1\right )+a b n x+b^2 (n+1) x^2\right )}{b^2 (n+1) (n+2)}+\frac{d^2 \left (\frac{b x}{a}+1\right )^{-n} \left (-6 a^4 \left (\left (\frac{b x}{a}+1\right )^n-1\right )+6 a^3 b n x \left (\frac{b x}{a}+1\right )^n-3 a^2 b^2 n (n+1) x^2 \left (\frac{b x}{a}+1\right )^n+b^4 \left (n^3+6 n^2+11 n+6\right ) x^4 \left (\frac{b x}{a}+1\right )^n+a b^3 n \left (n^2+3 n+2\right ) x^3 \left (\frac{b x}{a}+1\right )^n\right )}{b^4 (n+1) (n+2) (n+3) (n+4)}+\frac{c^2 \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^n*(c + d*x^2)^2)/x,x]
[Out]
(a + b*x)^n*((2*c*d*(a*b*n*x + b^2*(1 + n)*x^2 + a^2*(-1 + (1 + (b*x)/a)^(-n))))
/(b^2*(1 + n)*(2 + n)) + (d^2*(6*a^3*b*n*x*(1 + (b*x)/a)^n - 3*a^2*b^2*n*(1 + n)
*x^2*(1 + (b*x)/a)^n + a*b^3*n*(2 + 3*n + n^2)*x^3*(1 + (b*x)/a)^n + b^4*(6 + 11
*n + 6*n^2 + n^3)*x^4*(1 + (b*x)/a)^n - 6*a^4*(-1 + (1 + (b*x)/a)^n)))/(b^4*(1 +
n)*(2 + n)*(3 + n)*(4 + n)*(1 + (b*x)/a)^n) + (c^2*Hypergeometric2F1[-n, -n, 1
- n, -(a/(b*x))])/(n*(1 + a/(b*x))^n))
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Maple [F] time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( d{x}^{2}+c \right ) ^{2}}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x^2+c)^2/x,x)
[Out]
int((b*x+a)^n*(d*x^2+c)^2/x,x)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{2}{\left (b x + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*(b*x + a)^n/x,x, algorithm="maxima")
[Out]
integrate((d*x^2 + c)^2*(b*x + a)^n/x, x)
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )}{\left (b x + a\right )}^{n}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*(b*x + a)^n/x,x, algorithm="fricas")
[Out]
integral((d^2*x^4 + 2*c*d*x^2 + c^2)*(b*x + a)^n/x, x)
_______________________________________________________________________________________
Sympy [A] time = 15.5575, size = 1681, normalized size = 11.36 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**n*(d*x**2+c)**2/x,x)
[Out]
-b**n*c**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2
) - b**n*c**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n +
2) + 2*c*d*Piecewise((a**n*x**2/2, Eq(b, 0)), (a*log(a/b + x)/(a*b**2 + b**3*x)
+ b*x*log(a/b + x)/(a*b**2 + b**3*x) - b*x/(a*b**2 + b**3*x), Eq(n, -2)), (-a*lo
g(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**2*Piecewise((a**n*x**4/4, Eq(b, 0)), (6*a**3*log(
a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 2*a**3/
(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a**2*b*x*log(
a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b*
*2*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**
3) - 9*a*b**2*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3)
+ 6*b**3*x**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b
**7*x**3) - 9*b**3*x**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*
x**3), Eq(n, -4)), (-6*a**3*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2
) - 3*a**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*x*log(a/b + x)/(
2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*a*b**2*x**2*log(a/b + x)/(2*a**2*b**
4 + 4*a*b**5*x + 2*b**6*x**2) + 6*a*b**2*x**2/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6
*x**2) + 2*b**3*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2), Eq(n, -3)), (6*a*
*3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*
x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*x**2/(2*a*b**4 + 2*b**5*x) + b**
3*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*log(a/b + x)/b**4 + a**2*x/b**3
- a*x**2/(2*b**2) + x**3/(3*b), Eq(n, -1)), (-6*a**4*(a + b*x)**n/(b**4*n**4 +
10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*a**3*b*n*x*(a + b*x)**n/(
b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*n**
2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b*
*4) - 3*a**2*b**2*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 +
50*b**4*n + 24*b**4) + a*b**3*n**3*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3
+ 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*n**2*x**3*(a + b*x)**n/(b**4*n*
*4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b**3*n*x**3*(a + b
*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*n*
*3*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b
**4) + 6*b**4*n**2*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 +
50*b**4*n + 24*b**4) + 11*b**4*n*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 3
5*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*x**4*(a + b*x)**n/(b**4*n**4 + 10*b*
*4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4), True)) - b*b**n*c**2*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**2*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. \[ \int \frac{{\left (d x^{2} + c\right )}^{2}{\left (b x + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*(b*x + a)^n/x,x, algorithm="giac")
[Out]
integrate((d*x^2 + c)^2*(b*x + a)^n/x, x)