3.922 \(\int \frac{(a+b x)^n (c+d x)^3}{x^2} \, dx\)

Optimal. Leaf size=143 \[ -\frac{c^2 (a+b x)^{n+1} (3 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{(a+b x)^{n+1} \left (b c^2 (n+1) (a d+b c (n+2))+a d^2 x (a d-b c (n+4))\right )}{a b^2 (n+1) (n+2) x}+\frac{d (c+d x)^2 (a+b x)^{n+1}}{b (n+2) x} \]

[Out]

(d*(a + b*x)^(1 + n)*(c + d*x)^2)/(b*(2 + n)*x) - ((a + b*x)^(1 + n)*(b*c^2*(1 +
n)*(a*d + b*c*(2 + n)) + a*d^2*(a*d - b*c*(4 + n))*x))/(a*b^2*(1 + n)*(2 + n)*x
) - (c^2*(3*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 = 0.330754, antiderivative size = 143, 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 \[ -\frac{c^2 (a+b x)^{n+1} (3 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{(a+b x)^{n+1} \left (b c^2 (n+1) (a d+b c (n+2))+a d^2 x (a d-b c (n+4))\right )}{a b^2 (n+1) (n+2) x}+\frac{d (c+d x)^2 (a+b x)^{n+1}}{b (n+2) x} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^n*(c + d*x)^3)/x^2,x]

[Out]

(d*(a + b*x)^(1 + n)*(c + d*x)^2)/(b*(2 + n)*x) - ((a + b*x)^(1 + n)*(b*c^2*(1 +
n)*(a*d + b*c*(2 + n)) + a*d^2*(a*d - b*c*(4 + n))*x))/(a*b^2*(1 + n)*(2 + n)*x
) - (c^2*(3*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 = 26.4465, size = 121, normalized size = 0.85 \[ \frac{d \left (a + b x\right )^{n + 1} \left (c + d x\right )^{2}}{b x \left (n + 2\right )} - \frac{\left (a + b x\right )^{n + 1} \left (a d^{2} x \left (a d - b c \left (n + 4\right )\right ) + b c^{2} \left (n + 1\right ) \left (a d + b c \left (n + 2\right )\right )\right )}{a b^{2} x \left (n + 1\right ) \left (n + 2\right )} - \frac{c^{2} \left (a + b x\right )^{n + 1} \left (3 a d + b c n\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n*(d*x+c)**3/x**2,x)

[Out]

d*(a + b*x)**(n + 1)*(c + d*x)**2/(b*x*(n + 2)) - (a + b*x)**(n + 1)*(a*d**2*x*(
a*d - b*c*(n + 4)) + b*c**2*(n + 1)*(a*d + b*c*(n + 2)))/(a*b**2*x*(n + 1)*(n +
2)) - c**2*(a + b*x)**(n + 1)*(3*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 = 0.500982, size = 172, normalized size = 1.2 \[ (a+b x)^n \left (\frac{d^3 \left (a^2 \left (\left (\frac{b x}{a}+1\right )^{-n}-1\right )+a b n x+b^2 (n+1) x^2\right )}{b^2 (n+1) (n+2)}+\frac{c^3 \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (1-n,-n;2-n;-\frac{a}{b x}\right )}{(n-1) x}+\frac{3 c^2 d \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}+\frac{3 c d^2 (a+b x)}{b (n+1)}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^n*(c + d*x)^3)/x^2,x]

[Out]

(a + b*x)^n*((3*c*d^2*(a + b*x))/(b*(1 + n)) + (d^3*(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^3*Hypergeometric2F1[
1 - n, -n, 2 - n, -(a/(b*x))])/((-1 + n)*(1 + a/(b*x))^n*x) + (3*c^2*d*Hypergeom
etric2F1[-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. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{3}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)^3/x^2,x)

[Out]

int((b*x+a)^n*(d*x+c)^3/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)^3*(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. \[{\rm integral}\left (\frac{{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}{\left (b x + a\right )}^{n}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3*(b*x + a)^n/x^2,x, algorithm="fricas")

[Out]

integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*(b*x + a)^n/x^2, x)

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Sympy [A]  time = 16.3214, size = 770, normalized size = 5.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n*(d*x+c)**3/x**2,x)

[Out]

b**n*c**3*n**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x*gamma(
n + 2)) + b**n*c**3*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x
*gamma(n + 2)) - b**n*c**3*n*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - b**n*c
**3*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - 3*b**n*c**2*d*n*(a/b + x)**n*le
rchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - 3*b**n*c**2*d*(a/b + x)*
*n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + 3*c*d**2*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**3*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**3*n**2*(a/b + x)**n*lerc
hphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) + b*b**n*c**3*n*(a/b + x
)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b*b**n*c**3*n
*(a/b + x)**n*gamma(n + 1)/(a*gamma(n + 2)) - b*b**n*c**3*(a/b + x)**n*gamma(n +
1)/(a*gamma(n + 2)) - 3*b*b**n*c**2*d*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n
+ 1)*gamma(n + 1)/(a*gamma(n + 2)) - 3*b*b**n*c**2*d*x*(a/b + x)**n*lerchphi(1
+ b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b**2*b**n*c**3*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**3*n*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gam
ma(n + 1)/(a**2*x*gamma(n + 2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{3}{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3*(b*x + a)^n/x^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*(b*x + a)^n/x^2, x)