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

Optimal. Leaf size=88 \[ \frac{d (2 b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d^2 (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.108033, antiderivative size = 88, 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 \[ \frac{d (2 b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d^2 (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.3332, size = 71, normalized size = 0.81 \[ \frac{d^{2} \left (a + b x\right )^{n + 2}}{b^{2} \left (n + 2\right )} - \frac{d \left (a + b x\right )^{n + 1} \left (a d - 2 b c\right )}{b^{2} \left (n + 1\right )} - \frac{c^{2} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(a + b*x)**(n + 2)/(b**2*(n + 2)) - d*(a + b*x)**(n + 1)*(a*d - 2*b*c)/(b**
2*(n + 1)) - 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 = 0.31871, size = 119, normalized size = 1.35 \[ (a+b x)^n \left (\frac{d^2 \left (a^2 \left (\left (\frac{b x}{a}+1\right )^{-n}-1\right )+a b n x+b^2 (n+1) x^2\right )}{b^2 (n+1) (n+2)}+\frac{c^2 \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}+\frac{2 c d (a+b x)}{b (n+1)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{2}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)^2/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.1315, size = 386, normalized size = 4.39 \[ - \frac{b^{n} c^{2} n \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} - \frac{b^{n} c^{2} \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} + 2 c d \left (\begin{cases} a^{n} x & \text{for}\: b = 0 \\\frac{\begin{cases} \frac{\left (a + b x\right )^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (a + b x \right )} & \text{otherwise} \end{cases}}{b} & \text{otherwise} \end{cases}\right ) + d^{2} \left (\begin{cases} \frac{a^{n} x^{2}}{2} & \text{for}\: b = 0 \\\frac{a \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{a}{a b^{2} + b^{3} x} + \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text{for}\: n = -2 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{x}{b} & \text{for}\: n = -1 \\- \frac{a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text{otherwise} \end{cases}\right ) - \frac{b b^{n} c^{2} n x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac{b b^{n} c^{2} x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n*(d*x+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, Eq(b, 0)), (Piecewise(((a + b*x)**(n + 1)/(n + 1),
Ne(n, -1)), (log(a + b*x), True))/b, True)) + 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)) - 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))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{2}{\left (b x + a\right )}^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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