3.2.81 \(\int \frac {(a+b x)^n (c+d x^3)^2}{x} \, dx\) [181]
Optimal. Leaf size=209 \[ \frac {a^2 d \left (2 b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^6 (1+n)}-\frac {a d \left (4 b^3 c-5 a^3 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {10 a^2 d^2 (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^2 (a+b x)^{6+n}}{b^6 (6+n)}-\frac {c^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a (1+n)} \]
[Out]
a^2*d*(-a^3*d+2*b^3*c)*(b*x+a)^(1+n)/b^6/(1+n)-a*d*(-5*a^3*d+4*b^3*c)*(b*x+a)^(2+n)/b^6/(2+n)+2*d*(-5*a^3*d+b^
3*c)*(b*x+a)^(3+n)/b^6/(3+n)+10*a^2*d^2*(b*x+a)^(4+n)/b^6/(4+n)-5*a*d^2*(b*x+a)^(5+n)/b^6/(5+n)+d^2*(b*x+a)^(6
+n)/b^6/(6+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 = 0.09, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1634, 67}
\begin {gather*} -\frac {a d \left (4 b^3 c-5 a^3 d\right ) (a+b x)^{n+2}}{b^6 (n+2)}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {10 a^2 d^2 (a+b x)^{n+4}}{b^6 (n+4)}+\frac {a^2 d \left (2 b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^6 (n+1)}-\frac {5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d^2 (a+b x)^{n+6}}{b^6 (n+6)}-\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^3)^2)/x,x]
[Out]
(a^2*d*(2*b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^6*(1 + n)) - (a*d*(4*b^3*c - 5*a^3*d)*(a + b*x)^(2 + n))/(b^6*(
2 + n)) + (2*d*(b^3*c - 5*a^3*d)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (10*a^2*d^2*(a + b*x)^(4 + n))/(b^6*(4 + n
)) - (5*a*d^2*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^2*(a + b*x)^(6 + n))/(b^6*(6 + 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 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^3\right )^2}{x} \, dx &=\int \left (-\frac {a^2 d \left (-2 b^3 c+a^3 d\right ) (a+b x)^n}{b^5}+\frac {c^2 (a+b x)^n}{x}+\frac {a d \left (-4 b^3 c+5 a^3 d\right ) (a+b x)^{1+n}}{b^5}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^{2+n}}{b^5}+\frac {10 a^2 d^2 (a+b x)^{3+n}}{b^5}-\frac {5 a d^2 (a+b x)^{4+n}}{b^5}+\frac {d^2 (a+b x)^{5+n}}{b^5}\right ) \, dx\\ &=\frac {a^2 d \left (2 b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^6 (1+n)}-\frac {a d \left (4 b^3 c-5 a^3 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {10 a^2 d^2 (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^2 (a+b x)^{6+n}}{b^6 (6+n)}+c^2 \int \frac {(a+b x)^n}{x} \, dx\\ &=\frac {a^2 d \left (2 b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^6 (1+n)}-\frac {a d \left (4 b^3 c-5 a^3 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {10 a^2 d^2 (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^2 (a+b x)^{6+n}}{b^6 (6+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 = 0.16, size = 188, normalized size = 0.90 \begin {gather*} (a+b x)^{1+n} \left (\frac {a^2 d \left (2 b^3 c-a^3 d\right )}{b^6 (1+n)}+\frac {a d \left (-4 b^3 c+5 a^3 d\right ) (a+b x)}{b^6 (2+n)}+\frac {2 d \left (b^3 c-5 a^3 d\right ) (a+b x)^2}{b^6 (3+n)}+\frac {10 a^2 d^2 (a+b x)^3}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^4}{b^6 (5+n)}+\frac {d^2 (a+b x)^5}{b^6 (6+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^3)^2)/x,x]
[Out]
(a + b*x)^(1 + n)*((a^2*d*(2*b^3*c - a^3*d))/(b^6*(1 + n)) + (a*d*(-4*b^3*c + 5*a^3*d)*(a + b*x))/(b^6*(2 + n)
) + (2*d*(b^3*c - 5*a^3*d)*(a + b*x)^2)/(b^6*(3 + n)) + (10*a^2*d^2*(a + b*x)^3)/(b^6*(4 + n)) - (5*a*d^2*(a +
b*x)^4)/(b^6*(5 + n)) + (d^2*(a + b*x)^5)/(b^6*(6 + n)) - (c^2*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*x)/a
])/(a + a*n))
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n} \left (d \,x^{3}+c \right )^{2}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x^3+c)^2/x,x)
[Out]
int((b*x+a)^n*(d*x^3+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^3+c)^2/x,x, algorithm="maxima")
[Out]
integrate((d*x^3 + 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^3+c)^2/x,x, algorithm="fricas")
[Out]
integral((d^2*x^6 + 2*c*d*x^3 + 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. 4007 vs.
\(2 (187) = 374\).
time = 4.82, size = 4760, normalized size = 22.78 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**n*(d*x**3+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**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)) + d**2*Piecewise((a**n*x**6/6, Eq(b, 0)), (60*a**5*log(a/b +
x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**
5) + 137*a**5/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 6
0*b**11*x**5) + 300*a**4*b*x*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9
*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 625*a**4*b*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2
+ 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 600*a**3*b**2*x**2*log(a/b + x)/(60*a**5*b**6 + 300
*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 1100*a**3*b**2*x*
*2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**
5) + 600*a**2*b**3*x**3*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3
+ 300*a*b**10*x**4 + 60*b**11*x**5) + 900*a**2*b**3*x**3/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2
+ 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a*b**4*x**4*log(a/b + x)/(60*a**5*b**6 + 300*a
**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a*b**4*x**4/(60
*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 6
0*b**5*x**5*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**
10*x**4 + 60*b**11*x**5), Eq(n, -6)), (-60*a**5*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**
2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 125*a**5/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*
x**3 + 12*b**10*x**4) - 240*a**4*b*x*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b*
*9*x**3 + 12*b**10*x**4) - 440*a**4*b*x/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 +
12*b**10*x**4) - 360*a**3*b**2*x**2*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**
9*x**3 + 12*b**10*x**4) - 540*a**3*b**2*x**2/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x*
*3 + 12*b**10*x**4) - 240*a**2*b**3*x**3*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*
a*b**9*x**3 + 12*b**10*x**4) - 240*a**2*b**3*x**3/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b*
*9*x**3 + 12*b**10*x**4) - 60*a*b**4*x**4*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48
*a*b**9*x**3 + 12*b**10*x**4) + 12*b**5*x**5/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x*
*3 + 12*b**10*x**4), Eq(n, -5)), (60*a**5*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9
*x**3) + 110*a**5/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 180*a**4*b*x*log(a/b + x)/(6
*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 270*a**4*b*x/(6*a**3*b**6 + 18*a**2*b**7*x + 18*
a*b**8*x**2 + 6*b**9*x**3) + 180*a**3*b**2*x**2*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 +
6*b**9*x**3) + 180*a**3*b**2*x**2/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 60*a**2*b**3
*x**3*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) - 15*a*b**4*x**4/(6*a**3*b**6
+ 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 3*b**5*x**5/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2
+ 6*b**9*x**3), Eq(n, -4)), (-60*a**5*log(a/b + x)/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 90*a**5/(6*a**
2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 120*a**4*b*x*log(a/b + x)/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 12
0*a**4*b*x/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 60*a**3*b**2*x**2*log(a/b + x)/(6*a**2*b**6 + 12*a*b**7
*x + 6*b**8*x**2) + 20*a**2*b**3*x**3/(6*a**2*b...
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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^3+c)^2/x,x, algorithm="giac")
[Out]
integrate((d*x^3 + c)^2*(b*x + a)^n/x, x)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d\,x^3+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^3)^2*(a + b*x)^n)/x,x)
[Out]
int(((c + d*x^3)^2*(a + b*x)^n)/x, x)
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