3.10.29 \(\int \frac {(a+b x)^n (c+d x)^2}{x^4} \, dx\) [929]
Optimal. Leaf size=130 \[ -\frac {c^2 (a+b x)^{1+n}}{3 a x^3}-\frac {c (6 a d-b c (2-n)) (a+b x)^{1+n}}{6 a^2 x^2}+\frac {b \left (6 a^2 d^2-6 a b c d (1-n)+b^2 c^2 \left (2-3 n+n^2\right )\right ) (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {b x}{a}\right )}{6 a^4 (1+n)} \]
[Out]
-1/3*c^2*(b*x+a)^(1+n)/x^3/a-1/6*c*(6*a*d-b*c*(2-n))*(b*x+a)^(1+n)/a^2/x^2+1/6*b*(6*a^2*d^2-6*a*b*c*d*(1-n)+b^
2*c^2*(n^2-3*n+2))*(b*x+a)^(1+n)*hypergeom([2, 1+n],[2+n],1+b*x/a)/a^4/(1+n)
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Rubi [A]
time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 79, 67}
\begin {gather*} -\frac {c (a+b x)^{n+1} (6 a d-b c (2-n))}{6 a^2 x^2}+\frac {b (a+b x)^{n+1} \left (6 a^2 d^2-6 a b c d (1-n)+b^2 c^2 \left (n^2-3 n+2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac {b x}{a}+1\right )}{6 a^4 (n+1)}-\frac {c^2 (a+b x)^{n+1}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^n*(c + d*x)^2)/x^4,x]
[Out]
-1/3*(c^2*(a + b*x)^(1 + n))/(a*x^3) - (c*(6*a*d - b*c*(2 - n))*(a + b*x)^(1 + n))/(6*a^2*x^2) + (b*(6*a^2*d^2
- 6*a*b*c*d*(1 - n) + b^2*c^2*(2 - 3*n + n^2))*(a + b*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*x)
/a])/(6*a^4*(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 79
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || L
tQ[p, n]))))
Rule 91
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)^2}{x^4} \, dx &=-\frac {c^2 (a+b x)^{1+n}}{3 a x^3}+\frac {\int \frac {(a+b x)^n \left (c (6 a d-b c (2-n))+3 a d^2 x\right )}{x^3} \, dx}{3 a}\\ &=-\frac {c^2 (a+b x)^{1+n}}{3 a x^3}-\frac {c (6 a d-b c (2-n)) (a+b x)^{1+n}}{6 a^2 x^2}+\frac {1}{6} \left (6 d^2-\frac {b c (6 a d-b c (2-n)) (1-n)}{a^2}\right ) \int \frac {(a+b x)^n}{x^2} \, dx\\ &=-\frac {c^2 (a+b x)^{1+n}}{3 a x^3}-\frac {c (6 a d-b c (2-n)) (a+b x)^{1+n}}{6 a^2 x^2}+\frac {b \left (6 a^2 d^2-6 a b c d (1-n)+b^2 c^2 \left (2-3 n+n^2\right )\right ) (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {b x}{a}\right )}{6 a^4 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 105, normalized size = 0.81 \begin {gather*} \frac {(a+b x)^{1+n} \left (-a^2 c (1+n) (b c (-2+n) x+2 a (c+3 d x))+b \left (6 a^2 d^2+6 a b c d (-1+n)+b^2 c^2 \left (2-3 n+n^2\right )\right ) x^3 \, _2F_1\left (2,1+n;2+n;1+\frac {b x}{a}\right )\right )}{6 a^4 (1+n) x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^n*(c + d*x)^2)/x^4,x]
[Out]
((a + b*x)^(1 + n)*(-(a^2*c*(1 + n)*(b*c*(-2 + n)*x + 2*a*(c + 3*d*x))) + b*(6*a^2*d^2 + 6*a*b*c*d*(-1 + n) +
b^2*c^2*(2 - 3*n + n^2))*x^3*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*x)/a]))/(6*a^4*(1 + n)*x^3)
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{2}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x+c)^2/x^4,x)
[Out]
int((b*x+a)^n*(d*x+c)^2/x^4,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+c)^2/x^4,x, algorithm="maxima")
[Out]
integrate((d*x + c)^2*(b*x + a)^n/x^4, 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+c)^2/x^4,x, algorithm="fricas")
[Out]
integral((d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^n/x^4, x)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 5367 vs.
\(2 (114) = 228\).
time = 15.00, size = 5367, normalized size = 41.28 \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+c)**2/x**4,x)
[Out]
a**4*b**3*b**n*c**2*n**4*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a*
*6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - 2*a**4
*b**3*b**n*c**2*n**3*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b
*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - a**4*b**3*
b**n*c**2*n**3*(a/b + x)**n*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b
+ x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - a**4*b**3*b**n*c**2*n**2*(a/b + x)**n*lerchphi
(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**
2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) + 3*a**4*b**3*b**n*c**2*n**2*(a/b + x)**n*gamma(n + 1)
/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b
+ x)**3*gamma(n + 2)) + 2*a**4*b**3*b**n*c**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a*
*7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3
*gamma(n + 2)) - 2*a**4*b**3*b**n*c**2*n*(a/b + x)**n*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n
+ 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - 6*a**4*b**3*b**n*c**
2*(a/b + x)**n*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma
(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) + a**3*b**4*b**n*c**2*n**4*x*(a/b + x)**n*lerchphi(1 + b*x/a,
1, n + 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n +
2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - 2*a**3*b**4*b**n*c**2*n**3*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1
, n + 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2
) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - a**3*b**4*b**n*c**2*n**3*x*(a/b + x)**n*gamma(n + 1)/(12*a**7*gam
ma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma
(n + 2)) - a**3*b**4*b**n*c**2*n**2*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gamma(n
+ 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n +
2)) + 3*a**3*b**4*b**n*c**2*n**2*x*(a/b + x)**n*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2)
- 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) + 2*a**3*b**4*b**n*c**2*n*x
*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18
*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - 2*a**3*b**4*b**n*c**2*n*x*(a/b
+ x)**n*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2
) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) - 6*a**3*b**4*b**n*c**2*x*(a/b + x)**n*gamma(n + 1)/(12*a**7*gamma(
n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n
+ 2)) - 2*a**3*b**2*b**n*c*d*n**3*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2
) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + 2*a**3*b**2*b**n*c*d*n**2*(a/b + x)**n*
gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + 2*a**3
*b**2*b**n*c*d*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*ga
mma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - 2*a**3*b**2*b**n*c*d*n*(a/b + x)**n*gamma(n + 1)/(-2*a**
5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - 4*a**3*b**2*b**n*c*d*(a/b
+ x)**n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2))
- 3*a**2*b**5*b**n*c**2*n**4*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gam
ma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma
(n + 2)) + 6*a**2*b**5*b**n*c**2*n**3*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12
*a**7*gamma(n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)
**3*gamma(n + 2)) + 2*a**2*b**5*b**n*c**2*n**3*(a/b + x)**2*(a/b + x)**n*gamma(n + 1)/(12*a**7*gamma(n + 2) +
18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n + 2)) + 3
*a**2*b**5*b**n*c**2*n**2*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(12*a**7*gamma(
n + 2) + 18*a**6*b*x*gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(n + 2) + 6*a**4*b**3*(a/b + x)**3*gamma(n
+ 2)) - 5*a**2*b**5*b**n*c**2*n**2*(a/b + x)**2*(a/b + x)**n*gamma(n + 1)/(12*a**7*gamma(n + 2) + 18*a**6*b*x*
gamma(n + 2) - 18*a**5*b**2*(a/b + x)**2*gamma(...
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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+c)^2/x^4,x, algorithm="giac")
[Out]
integrate((d*x + c)^2*(b*x + a)^n/x^4, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a + b*x)^n*(c + d*x)^2)/x^4,x)
[Out]
int(((a + b*x)^n*(c + d*x)^2)/x^4, x)
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