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

Optimal. Leaf size=77 \[ -\frac{a d (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

-((a*d*(a + b*x)^(1 + n))/(b^2*(1 + n))) + (d*(a + b*x)^(2 + n))/(b^2*(2 + n)) -
(c*(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 = 0.120712, antiderivative size = 77, 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 \[ -\frac{a d (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((a*d*(a + b*x)^(1 + n))/(b^2*(1 + n))) + (d*(a + b*x)^(2 + n))/(b^2*(2 + n)) -
(c*(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 = 14.2698, size = 61, normalized size = 0.79 \[ - \frac{a d \left (a + b x\right )^{n + 1}}{b^{2} \left (n + 1\right )} + \frac{d \left (a + b x\right )^{n + 2}}{b^{2} \left (n + 2\right )} - \frac{c \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*d*(a + b*x)**(n + 1)/(b**2*(n + 1)) + d*(a + b*x)**(n + 2)/(b**2*(n + 2)) - c
*(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 = 0.29242, size = 98, normalized size = 1.27 \[ (a+b x)^n \left (\frac{d \left (a^2 \left (\left (\frac{b x}{a}+1\right )^{-n}-1\right )+a b n x+b^2 (n+1) x^2\right )}{b^2 (n+1) (n+2)}+\frac{c \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( d{x}^{2}+c \right ) }{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x^2+c)/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.02149, size = 347, normalized size = 4.51 \[ - \frac{b^{n} c n \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} - \frac{b^{n} c \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} + d \left (\begin{cases} \frac{a^{n} x^{2}}{2} & \text{for}\: b = 0 \\\frac{a \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} - \frac{b x}{a b^{2} + b^{3} x} & \text{for}\: n = -2 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{x}{b} & \text{for}\: n = -1 \\- \frac{a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text{otherwise} \end{cases}\right ) - \frac{b b^{n} c n x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac{b b^{n} c x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**n*c*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) -
b**n*c*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + 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*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*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*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. \[ \int \frac{{\left (d x^{2} + c\right )}{\left (b x + a\right )}^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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