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

Optimal. Leaf size=131 \[ \frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) (a+b x)^{n+1}}{b^3 (n+1)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d^3 (a+b x)^{n+3}}{b^3 (n+3)}-\frac {c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]

[Out]

d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1+n)/b^3/(1+n)+d^2*(-2*a*d+3*b*c)*(b*x+a)^(2+n)/b^3/(2+n)+d^3*(b*x+a)
^(3+n)/b^3/(3+n)-c^3*(b*x+a)^(1+n)*hypergeom([1, 1+n],[2+n],1+b*x/a)/a/(1+n)

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Rubi [A]  time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {88, 65} \[ \frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) (a+b x)^{n+1}}{b^3 (n+1)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d^3 (a+b x)^{n+3}}{b^3 (n+3)}-\frac {c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^n*(c + d*x)^3)/x,x]

[Out]

(d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*(a + b*x)^(1 + n))/(b^3*(1 + n)) + (d^2*(3*b*c - 2*a*d)*(a + b*x)^(2 + n)
)/(b^3*(2 + n)) + (d^3*(a + b*x)^(3 + n))/(b^3*(3 + n)) - (c^3*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2
+ n, 1 + (b*x)/a])/(a*(1 + n))

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) (a+b x)^n}{b^2}+\frac {c^3 (a+b x)^n}{x}+\frac {d^2 (3 b c-2 a d) (a+b x)^{1+n}}{b^2}+\frac {d^3 (a+b x)^{2+n}}{b^2}\right ) \, dx\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) (a+b x)^{1+n}}{b^3 (1+n)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d^3 (a+b x)^{3+n}}{b^3 (3+n)}+c^3 \int \frac {(a+b x)^n}{x} \, dx\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) (a+b x)^{1+n}}{b^3 (1+n)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d^3 (a+b x)^{3+n}}{b^3 (3+n)}-\frac {c^3 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a (1+n)}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 117, normalized size = 0.89 \[ (a+b x)^{n+1} \left (\frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (n+1)}+\frac {d^2 (a+b x) (3 b c-2 a d)}{b^3 (n+2)}+\frac {d^3 (a+b x)^2}{b^3 (n+3)}-\frac {c^3 \, _2F_1\left (1,n+1;n+2;\frac {a+b x}{a}\right )}{a n+a}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^(1 + n)*((d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2))/(b^3*(1 + n)) + (d^2*(3*b*c - 2*a*d)*(a + b*x))/(b^3*
(2 + n)) + (d^3*(a + b*x)^2)/(b^3*(3 + n)) - (c^3*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*x)/a])/(a + a*n))

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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} {\left (b x + a\right )}^{n}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(d*x+c)^3/x,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, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{3} {\left (b x + a\right )}^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{3} \left (b x +a \right )^{n}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3 \, {\left (b x + a\right )}^{n + 1} c^{2} d}{b {\left (n + 1\right )}} + \int \frac {{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3}\right )} {\left (b x + a\right )}^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b*x + a)^(n + 1)*c^2*d/(b*(n + 1)) + integrate((d^3*x^3 + 3*c*d^2*x^2 + c^3)*(b*x + a)^n/x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^3}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^n*(c + d*x)^3)/x,x)

[Out]

int(((a + b*x)^n*(c + d*x)^3)/x, x)

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sympy [B]  time = 10.03, size = 993, normalized size = 7.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**n*c**3*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - b**n*c**3*(a/b + x)**n*ler
chphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + 3*c**2*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)) + 3*c*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)) + d**3*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*log(
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**3*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*ga
mma(n + 1)/(a*gamma(n + 2)) - b*b**n*c**3*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n
+ 2))

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