3.154 \(\int \frac{(a+b x)^n \left (c+d x^3\right )}{x} \, dx\)

Optimal. Leaf size=99 \[ \frac{a^2 d (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

(a^2*d*(a + b*x)^(1 + n))/(b^3*(1 + n)) - (2*a*d*(a + b*x)^(2 + n))/(b^3*(2 + n)
) + (d*(a + b*x)^(3 + n))/(b^3*(3 + 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 = 0.115322, antiderivative size = 99, 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 \[ \frac{a^2 d (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^n*(c + d*x^3))/x,x]

[Out]

(a^2*d*(a + b*x)^(1 + n))/(b^3*(1 + n)) - (2*a*d*(a + b*x)^(2 + n))/(b^3*(2 + n)
) + (d*(a + b*x)^(3 + n))/(b^3*(3 + 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 = 23.2067, size = 83, normalized size = 0.84 \[ \frac{a^{2} d \left (a + b x\right )^{n + 1}}{b^{3} \left (n + 1\right )} - \frac{2 a d \left (a + b x\right )^{n + 2}}{b^{3} \left (n + 2\right )} + \frac{d \left (a + b x\right )^{n + 3}}{b^{3} \left (n + 3\right )} - \frac{c \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n*(d*x**3+c)/x,x)

[Out]

a**2*d*(a + b*x)**(n + 1)/(b**3*(n + 1)) - 2*a*d*(a + b*x)**(n + 2)/(b**3*(n + 2
)) + d*(a + b*x)**(n + 3)/(b**3*(n + 3)) - 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 = 0.251355, size = 125, normalized size = 1.26 \[ (a+b x)^n \left (\frac{d \left (a^3 \left (2-2 \left (\frac{b x}{a}+1\right )^{-n}\right )-2 a^2 b n x+a b^2 n (n+1) x^2+b^3 \left (n^2+3 n+2\right ) x^3\right )}{b^3 \left (n^3+6 n^2+11 n+6\right )}+\frac{c \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^n*(c + d*x^3))/x,x]

[Out]

(a + b*x)^n*((d*(-2*a^2*b*n*x + a*b^2*n*(1 + n)*x^2 + b^3*(2 + 3*n + n^2)*x^3 +
a^3*(2 - 2/(1 + (b*x)/a)^n)))/(b^3*(6 + 11*n + 6*n^2 + n^3)) + (c*Hypergeometric
2F1[-n, -n, 1 - n, -(a/(b*x))])/(n*(1 + a/(b*x))^n))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( d{x}^{3}+c \right ) }{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x^3+c)/x,x)

[Out]

int((b*x+a)^n*(d*x^3+c)/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}{\left (b x + a\right )}^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)*(b*x + a)^n/x,x, algorithm="maxima")

[Out]

integrate((d*x^3 + c)*(b*x + a)^n/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{3} + c\right )}{\left (b x + a\right )}^{n}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)*(b*x + a)^n/x,x, algorithm="fricas")

[Out]

integral((d*x^3 + c)*(b*x + a)^n/x, x)

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Sympy [A]  time = 11.0848, size = 741, normalized size = 7.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n*(d*x**3+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**3/3, Eq(b, 0)), (2*a**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4
*x + 2*b**5*x**2) + 3*a**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*x*lo
g(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*x/(2*a**2*b**3 + 4*a
*b**4*x + 2*b**5*x**2) + 2*b**2*x**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*
b**5*x**2), Eq(n, -3)), (-2*a**2*log(a/b + x)/(a*b**3 + b**4*x) - 2*a**2/(a*b**3
+ b**4*x) - 2*a*b*x*log(a/b + x)/(a*b**3 + b**4*x) + b**2*x**2/(a*b**3 + b**4*x
), Eq(n, -2)), (a**2*log(a/b + x)/b**3 - a*x/b**2 + x**2/(2*b), Eq(n, -1)), (2*a
**3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 2*a**2*b*n*x*(
a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*n**2*x**2*(a
+ b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*n*x**2*(a + b
*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + b**3*n**2*x**3*(a + b*x)
**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 3*b**3*n*x**3*(a + b*x)**n/
(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 2*b**3*x**3*(a + b*x)**n/(b**3*
n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3), True)) - b*b**n*c*n*x*(a/b + x)**n*ler
chphi(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. \[ \int \frac{{\left (d x^{3} + c\right )}{\left (b x + a\right )}^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)*(b*x + a)^n/x,x, algorithm="giac")

[Out]

integrate((d*x^3 + c)*(b*x + a)^n/x, x)