3.915 \(\int \frac{(a+b x)^n (c+d x)^2}{x^2} \, dx\)

Optimal. Leaf size=87 \[ -\frac{c (a+b x)^{n+1} (2 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{c^2 (a+b x)^{n+1}}{a x}+\frac{d^2 (a+b x)^{n+1}}{b (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.115406, antiderivative size = 87, 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 \[ -\frac{c (a+b x)^{n+1} (2 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{c^2 (a+b x)^{n+1}}{a x}+\frac{d^2 (a+b x)^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^n*(c + d*x)^2)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.4069, size = 70, normalized size = 0.8 \[ \frac{d^{2} \left (a + b x\right )^{n + 1}}{b \left (n + 1\right )} - \frac{c^{2} \left (a + b x\right )^{n + 1}}{a x} - \frac{c \left (a + b x\right )^{n + 1} \left (2 a d + b c n\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n*(d*x+c)**2/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.308175, size = 117, normalized size = 1.34 \[ (a+b x)^n \left (\frac{c^2 \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (1-n,-n;2-n;-\frac{a}{b x}\right )}{(n-1) x}+\frac{2 c d \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}+\frac{d (a d+b d x)}{b n+b}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^n*(c + d*x)^2)/x^2,x]

[Out]

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

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{2}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)^2/x^2,x)

[Out]

int((b*x+a)^n*(d*x+c)^2/x^2,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)^2*(b*x + a)^n/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^2*(b*x + a)^n/x^2,x, algorithm="fricas")

[Out]

integral((d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^n/x^2, x)

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Sympy [A]  time = 13.4361, size = 554, normalized size = 6.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n*(d*x+c)**2/x**2,x)

[Out]

b**n*c**2*n**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x*gamma(
n + 2)) + b**n*c**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x
*gamma(n + 2)) - b**n*c**2*n*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - b**n*c
**2*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - 2*b**n*c*d*n*(a/b + x)**n*lerch
phi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - 2*b**n*c*d*(a/b + x)**n*ler
chphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + d**2*Piecewise((a**n*x, E
q(b, 0)), (Piecewise(((a + b*x)**(n + 1)/(n + 1), Ne(n, -1)), (log(a + b*x), Tru
e))/b, True)) + b*b**n*c**2*n**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamm
a(n + 1)/(a*gamma(n + 2)) + b*b**n*c**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n
+ 1)*gamma(n + 1)/(a*gamma(n + 2)) - b*b**n*c**2*n*(a/b + x)**n*gamma(n + 1)/(a*
gamma(n + 2)) - b*b**n*c**2*(a/b + x)**n*gamma(n + 1)/(a*gamma(n + 2)) - 2*b*b**
n*c*d*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2
)) - 2*b*b**n*c*d*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*g
amma(n + 2)) - b**2*b**n*c**2*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)) - b**2*b**n*c**2*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. \[ \int \frac{{\left (d x + c\right )}^{2}{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^2*(b*x + a)^n/x^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^2*(b*x + a)^n/x^2, x)