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

Optimal. Leaf size=62 \[ -\frac{(a+b x)^{n+1} (a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{c (a+b x)^{n+1}}{a x} \]

[Out]

-((c*(a + b*x)^(1 + n))/(a*x)) - ((a*d + b*c*n)*(a + b*x)^(1 + n)*Hypergeometric
2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a^2*(1 + n))

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Rubi [A]  time = 0.0603238, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{(a+b x)^{n+1} (a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{c (a+b x)^{n+1}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-((c*(a + b*x)^(1 + n))/(a*x)) - ((a*d + b*c*n)*(a + b*x)^(1 + n)*Hypergeometric
2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a^2*(1 + n))

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Rubi in Sympy [A]  time = 6.89646, size = 49, normalized size = 0.79 \[ - \frac{c \left (a + b x\right )^{n + 1}}{a x} - \frac{\left (a + b x\right )^{n + 1} \left (a d + b c n\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*(a + b*x)**(n + 1)/(a*x) - (a + b*x)**(n + 1)*(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 = 0.0500565, size = 87, normalized size = 1.4 \[ \frac{\left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \left (c n \, _2F_1\left (1-n,-n;2-n;-\frac{a}{b x}\right )+d (n-1) x \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )\right )}{(n-1) n x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) }{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 15.6143, size = 493, normalized size = 7.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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