3.4.58 \(\int \frac {(a+b x)^n (c+d x^2)^2}{x} \, dx\) [358]

Optimal. Leaf size=148 \[ -\frac {a d \left (2 b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^4 (1+n)}+\frac {d \left (2 b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^4 (2+n)}-\frac {3 a d^2 (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d^2 (a+b x)^{4+n}}{b^4 (4+n)}-\frac {c^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a (1+n)} \]

[Out]

-a*d*(a^2*d+2*b^2*c)*(b*x+a)^(1+n)/b^4/(1+n)+d*(3*a^2*d+2*b^2*c)*(b*x+a)^(2+n)/b^4/(2+n)-3*a*d^2*(b*x+a)^(3+n)
/b^4/(3+n)+d^2*(b*x+a)^(4+n)/b^4/(4+n)-c^2*(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 = 0.14, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {966, 1634, 67} \begin {gather*} -\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)} \end {gather*}

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 67

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

Rule 966

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 1634

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]

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*}

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Mathematica [A]
time = 0.14, size = 132, normalized size = 0.89 \begin {gather*} (a+b x)^{1+n} \left (-\frac {a d \left (2 b^2 c+a^2 d\right )}{b^4 (1+n)}+\frac {d \left (2 b^2 c+3 a^2 d\right ) (a+b x)}{b^4 (2+n)}-\frac {3 a d^2 (a+b x)^2}{b^4 (3+n)}+\frac {d^2 (a+b x)^3}{b^4 (4+n)}-\frac {c^2 \, _2F_1\left (1,1+n;2+n;\frac {a+b x}{a}\right )}{a+a n}\right ) \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n} \left (d \,x^{2}+c \right )^{2}}{x}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1321 vs. \(2 (129) = 258\).
time = 3.71, size = 1678, normalized size = 11.34 \begin {gather*} - \frac {b^{n} c^{2} n \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} - \frac {b^{n} c^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} + 2 c d \left (\begin {cases} \frac {a^{n} x^{2}}{2} & \text {for}\: b = 0 \\\frac {a \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {a}{a b^{2} + b^{3} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: n = -2 \\- \frac {a \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {x}{b} & \text {for}\: n = -1 \\- \frac {a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text {otherwise} \end {cases}\right ) + d^{2} \left (\begin {cases} \frac {a^{n} x^{4}}{4} & \text {for}\: b = 0 \\\frac {6 a^{3} \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {11 a^{3}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {18 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {27 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}} + \frac {18 a b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {18 a b^{2} x^{2}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {6 b^{3} x^{3} \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} & \text {for}\: n = -4 \\- \frac {6 a^{3} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {9 a^{3}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {12 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {12 a^{2} b x}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {6 a b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {2 b^{3} x^{3}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} & \text {for}\: n = -3 \\\frac {6 a^{3} \log {\left (\frac {a}{b} + x \right )}}{2 a b^{4} + 2 b^{5} x} + \frac {6 a^{3}}{2 a b^{4} + 2 b^{5} x} + \frac {6 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{2 a b^{4} + 2 b^{5} x} - \frac {3 a b^{2} x^{2}}{2 a b^{4} + 2 b^{5} x} + \frac {b^{3} x^{3}}{2 a b^{4} + 2 b^{5} x} & \text {for}\: n = -2 \\- \frac {a^{3} \log {\left (\frac {a}{b} + x \right )}}{b^{4}} + \frac {a^{2} x}{b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{3}}{3 b} & \text {for}\: n = -1 \\- \frac {6 a^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 a^{3} b n x \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} - \frac {3 a^{2} b^{2} n^{2} x^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} - \frac {3 a^{2} b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {a b^{3} n^{3} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {3 a b^{3} n^{2} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {2 a b^{3} n x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {b^{4} n^{3} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 b^{4} n^{2} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {11 b^{4} n x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 b^{4} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} & \text {otherwise} \end {cases}\right ) - \frac {b b^{n} c^{2} n x \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b b^{n} c^{2} x \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \end {gather*}

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) + 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**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) + 11*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) + 27*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) + 18*a*b**2*x**2/(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), 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*lerchph
i(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
)*gamma(n + 1)/(a*gamma(n + 2))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^2\,{\left (a+b\,x\right )}^n}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c + d*x^2)^2*(a + b*x)^n)/x,x)

[Out]

int(((c + d*x^2)^2*(a + b*x)^n)/x, x)

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