3.358 \(\int \frac{(a+b x)^n (c+d x^2)^2}{x} \, dx\)
Optimal. Leaf size=148 \[ -\frac{a d \left (a^2 d+2 b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{d \left (3 a^2 d+2 b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d^2 (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d^2 (a+b x)^{n+4}}{b^4 (n+4)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]
[Out]
-((a*d*(2*b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^4*(1 + n))) + (d*(2*b^2*c + 3*a^2*d)*(a + b*x)^(2 + n))/(b^4*(2
+ n)) - (3*a*d^2*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n)) - (c^2*(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 = 0.209861, antiderivative size = 148, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used =
{952, 1620, 65} \[ -\frac{a d \left (a^2 d+2 b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{d \left (3 a^2 d+2 b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d^2 (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d^2 (a+b x)^{n+4}}{b^4 (n+4)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^n*(c + d*x^2)^2)/x,x]
[Out]
-((a*d*(2*b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^4*(1 + n))) + (d*(2*b^2*c + 3*a^2*d)*(a + b*x)^(2 + n))/(b^4*(2
+ n)) - (3*a*d^2*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n)) - (c^2*(a + b*x)^(1
+ n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*(1 + n))
Rule 952
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(c^p*(d
+ e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m + n + 2*p + 1)),
Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c
^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0]
&& NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[n] || !IntegerQ[m])
Rule 1620
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]
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])
Rubi steps
\begin{align*} \int \frac{(a+b x)^n \left (c+d x^2\right )^2}{x} \, dx &=\frac{d^2 (a+b x)^{4+n}}{b^4 (4+n)}+\frac{\int \frac{(a+b x)^n \left (b^4 c^2 (4+n)-a^3 b d^2 (4+n) x+b^2 d \left (2 b^2 c-3 a^2 d\right ) (4+n) x^2-3 a b^3 d^2 (4+n) x^3\right )}{x} \, dx}{b^4 (4+n)}\\ &=\frac{d^2 (a+b x)^{4+n}}{b^4 (4+n)}+\frac{\int \left (-a b d \left (2 b^2 c+a^2 d\right ) (4+n) (a+b x)^n+\frac{\left (4 b^4 c^2+b^4 c^2 n\right ) (a+b x)^n}{x}+b d \left (2 b^2 c+3 a^2 d\right ) (4+n) (a+b x)^{1+n}-3 a b d^2 (4+n) (a+b x)^{2+n}\right ) \, dx}{b^4 (4+n)}\\ &=-\frac{a d \left (2 b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^4 (1+n)}+\frac{d \left (2 b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^4 (2+n)}-\frac{3 a d^2 (a+b x)^{3+n}}{b^4 (3+n)}+\frac{d^2 (a+b x)^{4+n}}{b^4 (4+n)}+c^2 \int \frac{(a+b x)^n}{x} \, dx\\ &=-\frac{a d \left (2 b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^4 (1+n)}+\frac{d \left (2 b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^4 (2+n)}-\frac{3 a d^2 (a+b x)^{3+n}}{b^4 (3+n)}+\frac{d^2 (a+b x)^{4+n}}{b^4 (4+n)}-\frac{c^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{a (1+n)}\\ \end{align*}
Mathematica [A] time = 0.142298, size = 132, normalized size = 0.89 \[ (a+b x)^{n+1} \left (\frac{d (a+b x) \left (3 a^2 d+2 b^2 c\right )}{b^4 (n+2)}-\frac{a d \left (a^2 d+2 b^2 c\right )}{b^4 (n+1)}+\frac{d^2 (a+b x)^3}{b^4 (n+4)}-\frac{3 a d^2 (a+b x)^2}{b^4 (n+3)}-\frac{c^2 \, _2F_1\left (1,n+1;n+2;\frac{a+b x}{a}\right )}{a n+a}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^n*(c + d*x^2)^2)/x,x]
[Out]
(a + b*x)^(1 + n)*(-((a*d*(2*b^2*c + a^2*d))/(b^4*(1 + n))) + (d*(2*b^2*c + 3*a^2*d)*(a + b*x))/(b^4*(2 + n))
- (3*a*d^2*(a + b*x)^2)/(b^4*(3 + n)) + (d^2*(a + b*x)^3)/(b^4*(4 + n)) - (c^2*Hypergeometric2F1[1, 1 + n, 2 +
n, (a + b*x)/a])/(a + a*n))
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Maple [F] time = 0.53, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( d{x}^{2}+c \right ) ^{2}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x^2+c)^2/x,x)
[Out]
int((b*x+a)^n*(d*x^2+c)^2/x,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{2}{\left (b x + a\right )}^{n}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^2/x,x, algorithm="maxima")
[Out]
integrate((d*x^2 + c)^2*(b*x + a)^n/x, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )}{\left (b x + a\right )}^{n}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^2/x,x, algorithm="fricas")
[Out]
integral((d^2*x^4 + 2*c*d*x^2 + c^2)*(b*x + a)^n/x, x)
________________________________________________________________________________________
Sympy [B] time = 11.2243, size = 1678, normalized size = 11.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**n*(d*x**2+c)**2/x,x)
[Out]
-b**n*c**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - b**n*c**2*(a/b + x)**n*ler
chphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + 2*c*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)) + d**2*Piecewise((a**n*x**4/4, Eq(b, 0)), (6*a**3*log(a/b + x)/(6*a**3*b**
4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 5*a**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6
*b**7*x**3) + 18*a**2*b*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 9*a**2*
b*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*x**2*log(a/b + x)/(6*a**3*b**4 +
18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3*x**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*
a*b**6*x**2 + 6*b**7*x**3) - 6*b**3*x**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq(n,
-4)), (-6*a**3*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b*
*6*x**2) - 12*a**2*b*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*x/(2*a**2*b**4 + 4*a*
b**5*x + 2*b**6*x**2) - 6*a*b**2*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*x**3/(2*a
**2*b**4 + 4*a*b**5*x + 2*b**6*x**2), Eq(n, -3)), (6*a**3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3/(2*a*b**
4 + 2*b**5*x) + 6*a**2*b*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*x**2/(2*a*b**4 + 2*b**5*x) + b**3*x**
3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*log(a/b + x)/b**4 + a**2*x/b**3 - a*x**2/(2*b**2) + x**3/(3*b), Eq
(n, -1)), (-6*a**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*a**3*b*n*x
*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*n**2*x**2*(a + b*x
)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*n*x**2*(a + b*x)**n/(b**4*n
**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*n**3*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*
n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*n**2*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b*
*4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b**3*n*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b*
*4*n + 24*b**4) + b**4*n**3*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4)
+ 6*b**4*n**2*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 11*b**4*n*x*
*4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*x**4*(a + b*x)**n/(b*
*4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4), True)) - b*b**n*c**2*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**2*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*g
amma(n + 1)/(a*gamma(n + 2))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{2}{\left (b x + a\right )}^{n}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^2/x,x, algorithm="giac")
[Out]
integrate((d*x^2 + c)^2*(b*x + a)^n/x, x)