3.399 \(\int x^5 (d+e x)^3 (a+b x^2)^p \, dx\)
Optimal. Leaf size=247 \[ \frac {a^2 d \left (b d^2-3 a e^2\right ) \left (a+b x^2\right )^{p+1}}{2 b^4 (p+1)}-\frac {a d \left (2 b d^2-9 a e^2\right ) \left (a+b x^2\right )^{p+2}}{2 b^4 (p+2)}+\frac {d \left (b d^2-9 a e^2\right ) \left (a+b x^2\right )^{p+3}}{2 b^4 (p+3)}+\frac {3 d e^2 \left (a+b x^2\right )^{p+4}}{2 b^4 (p+4)}-\frac {e x^7 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (7 a e^2-3 b d^2 (2 p+9)\right ) \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};-\frac {b x^2}{a}\right )}{7 b (2 p+9)}+\frac {e^3 x^7 \left (a+b x^2\right )^{p+1}}{b (2 p+9)} \]
[Out]
1/2*a^2*d*(-3*a*e^2+b*d^2)*(b*x^2+a)^(1+p)/b^4/(1+p)+e^3*x^7*(b*x^2+a)^(1+p)/b/(9+2*p)-1/2*a*d*(-9*a*e^2+2*b*d
^2)*(b*x^2+a)^(2+p)/b^4/(2+p)+1/2*d*(-9*a*e^2+b*d^2)*(b*x^2+a)^(3+p)/b^4/(3+p)+3/2*d*e^2*(b*x^2+a)^(4+p)/b^4/(
4+p)-1/7*e*(7*a*e^2-3*b*d^2*(9+2*p))*x^7*(b*x^2+a)^p*hypergeom([7/2, -p],[9/2],-b*x^2/a)/b/(9+2*p)/((1+b*x^2/a
)^p)
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Rubi [A] time = 0.25, antiderivative size = 241, normalized size of antiderivative = 0.98,
number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used =
{1652, 446, 77, 459, 365, 364} \[ \frac {a^2 d \left (b d^2-3 a e^2\right ) \left (a+b x^2\right )^{p+1}}{2 b^4 (p+1)}-\frac {a d \left (2 b d^2-9 a e^2\right ) \left (a+b x^2\right )^{p+2}}{2 b^4 (p+2)}+\frac {d \left (b d^2-9 a e^2\right ) \left (a+b x^2\right )^{p+3}}{2 b^4 (p+3)}+\frac {3 d e^2 \left (a+b x^2\right )^{p+4}}{2 b^4 (p+4)}+\frac {1}{7} e x^7 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 d^2-\frac {7 a e^2}{2 b p+9 b}\right ) \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};-\frac {b x^2}{a}\right )+\frac {e^3 x^7 \left (a+b x^2\right )^{p+1}}{b (2 p+9)} \]
Antiderivative was successfully verified.
[In]
Int[x^5*(d + e*x)^3*(a + b*x^2)^p,x]
[Out]
(a^2*d*(b*d^2 - 3*a*e^2)*(a + b*x^2)^(1 + p))/(2*b^4*(1 + p)) + (e^3*x^7*(a + b*x^2)^(1 + p))/(b*(9 + 2*p)) -
(a*d*(2*b*d^2 - 9*a*e^2)*(a + b*x^2)^(2 + p))/(2*b^4*(2 + p)) + (d*(b*d^2 - 9*a*e^2)*(a + b*x^2)^(3 + p))/(2*b
^4*(3 + p)) + (3*d*e^2*(a + b*x^2)^(4 + p))/(2*b^4*(4 + p)) + (e*(3*d^2 - (7*a*e^2)/(9*b + 2*b*p))*x^7*(a + b*
x^2)^p*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(7*(1 + (b*x^2)/a)^p)
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rule 364
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rule 365
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[
p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Rule 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 459
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]
Rule 1652
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[x^m*Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !PolyQ[Pq, x^2] && IGtQ[m, -2] && !
IntegerQ[2*p]
Rubi steps
\begin {align*} \int x^5 (d+e x)^3 \left (a+b x^2\right )^p \, dx &=\int x^5 \left (a+b x^2\right )^p \left (d^3+3 d e^2 x^2\right ) \, dx+\int x^6 \left (a+b x^2\right )^p \left (3 d^2 e+e^3 x^2\right ) \, dx\\ &=\frac {e^3 x^7 \left (a+b x^2\right )^{1+p}}{b (9+2 p)}+\frac {1}{2} \operatorname {Subst}\left (\int x^2 (a+b x)^p \left (d^3+3 d e^2 x\right ) \, dx,x,x^2\right )+\left (e \left (3 d^2-\frac {7 a e^2}{9 b+2 b p}\right )\right ) \int x^6 \left (a+b x^2\right )^p \, dx\\ &=\frac {e^3 x^7 \left (a+b x^2\right )^{1+p}}{b (9+2 p)}+\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^2 d \left (-b d^2+3 a e^2\right ) (a+b x)^p}{b^3}+\frac {a d \left (-2 b d^2+9 a e^2\right ) (a+b x)^{1+p}}{b^3}+\frac {\left (b d^3-9 a d e^2\right ) (a+b x)^{2+p}}{b^3}+\frac {3 d e^2 (a+b x)^{3+p}}{b^3}\right ) \, dx,x,x^2\right )+\left (e \left (3 d^2-\frac {7 a e^2}{9 b+2 b p}\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int x^6 \left (1+\frac {b x^2}{a}\right )^p \, dx\\ &=\frac {a^2 d \left (b d^2-3 a e^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^4 (1+p)}+\frac {e^3 x^7 \left (a+b x^2\right )^{1+p}}{b (9+2 p)}-\frac {a d \left (2 b d^2-9 a e^2\right ) \left (a+b x^2\right )^{2+p}}{2 b^4 (2+p)}+\frac {d \left (b d^2-9 a e^2\right ) \left (a+b x^2\right )^{3+p}}{2 b^4 (3+p)}+\frac {3 d e^2 \left (a+b x^2\right )^{4+p}}{2 b^4 (4+p)}+\frac {1}{7} e \left (3 d^2-\frac {7 a e^2}{9 b+2 b p}\right ) x^7 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};-\frac {b x^2}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 249, normalized size = 1.01 \[ \frac {1}{126} \left (a+b x^2\right )^p \left (\frac {63 d^3 \left (a+b x^2\right ) \left (2 a^2-2 a b (p+1) x^2+b^2 \left (p^2+3 p+2\right ) x^4\right )}{b^3 (p+1) (p+2) (p+3)}+\frac {189 d e^2 \left (a+b x^2\right ) \left (-6 a^3+6 a^2 b (p+1) x^2-3 a b^2 \left (p^2+3 p+2\right ) x^4+b^3 \left (p^3+6 p^2+11 p+6\right ) x^6\right )}{b^4 (p+1) (p+2) (p+3) (p+4)}+54 d^2 e x^7 \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};-\frac {b x^2}{a}\right )+14 e^3 x^9 \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {9}{2},-p;\frac {11}{2};-\frac {b x^2}{a}\right )\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^5*(d + e*x)^3*(a + b*x^2)^p,x]
[Out]
((a + b*x^2)^p*((63*d^3*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p
)*(3 + p)) + (189*d*e^2*(a + b*x^2)*(-6*a^3 + 6*a^2*b*(1 + p)*x^2 - 3*a*b^2*(2 + 3*p + p^2)*x^4 + b^3*(6 + 11*
p + 6*p^2 + p^3)*x^6))/(b^4*(1 + p)*(2 + p)*(3 + p)*(4 + p)) + (54*d^2*e*x^7*Hypergeometric2F1[7/2, -p, 9/2, -
((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (14*e^3*x^9*Hypergeometric2F1[9/2, -p, 11/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^
p))/126
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{3} x^{8} + 3 \, d e^{2} x^{7} + 3 \, d^{2} e x^{6} + d^{3} x^{5}\right )} {\left (b x^{2} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(e*x+d)^3*(b*x^2+a)^p,x, algorithm="fricas")
[Out]
integral((e^3*x^8 + 3*d*e^2*x^7 + 3*d^2*e*x^6 + d^3*x^5)*(b*x^2 + a)^p, x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{3} {\left (b x^{2} + a\right )}^{p} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(e*x+d)^3*(b*x^2+a)^p,x, algorithm="giac")
[Out]
integrate((e*x + d)^3*(b*x^2 + a)^p*x^5, x)
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{3} x^{5} \left (b \,x^{2}+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5*(e*x+d)^3*(b*x^2+a)^p,x)
[Out]
int(x^5*(e*x+d)^3*(b*x^2+a)^p,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} x^{6} + {\left (p^{2} + p\right )} a b^{2} x^{4} - 2 \, a^{2} b p x^{2} + 2 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{p} d^{3}}{2 \, {\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3}} + \int {\left (e^{3} x^{8} + 3 \, d e^{2} x^{7} + 3 \, d^{2} e x^{6}\right )} {\left (b x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^5*(e*x+d)^3*(b*x^2+a)^p,x, algorithm="maxima")
[Out]
1/2*((p^2 + 3*p + 2)*b^3*x^6 + (p^2 + p)*a*b^2*x^4 - 2*a^2*b*p*x^2 + 2*a^3)*(b*x^2 + a)^p*d^3/((p^3 + 6*p^2 +
11*p + 6)*b^3) + integrate((e^3*x^8 + 3*d*e^2*x^7 + 3*d^2*e*x^6)*(b*x^2 + a)^p, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^5\,{\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5*(a + b*x^2)^p*(d + e*x)^3,x)
[Out]
int(x^5*(a + b*x^2)^p*(d + e*x)^3, x)
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sympy [C] time = 74.61, size = 3082, normalized size = 12.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**5*(e*x+d)**3*(b*x**2+a)**p,x)
[Out]
3*a**p*d**2*e*x**7*hyper((7/2, -p), (9/2,), b*x**2*exp_polar(I*pi)/a)/7 + a**p*e**3*x**9*hyper((9/2, -p), (11/
2,), b*x**2*exp_polar(I*pi)/a)/9 + d**3*Piecewise((a**p*x**6/6, Eq(b, 0)), (2*a**2*log(-I*sqrt(a)*sqrt(1/b) +
x)/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 2*a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x
**2 + 4*b**5*x**4) + 3*a**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 4*a*b*x**2*log(-I*sqrt(a)*sqrt(1/b)
+ x)/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 4*a*b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*
b**4*x**2 + 4*b**5*x**4) + 4*a*b*x**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 2*b**2*x**4*log(-I*sqrt(a)
*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 2*b**2*x**4*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2
*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4), Eq(p, -3)), (-2*a**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**4*x*
*2) - 2*a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**4*x**2) - 2*a**2/(2*a*b**3 + 2*b**4*x**2) - 2*a*b*x
**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**4*x**2) - 2*a*b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3
+ 2*b**4*x**2) + b**2*x**4/(2*a*b**3 + 2*b**4*x**2), Eq(p, -2)), (a**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*b**3)
+ a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*b**3) - a*x**2/(2*b**2) + x**4/(4*b), Eq(p, -1)), (2*a**3*(a + b*x**2)
**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) - 2*a**2*b*p*x**2*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**
3*p**2 + 22*b**3*p + 12*b**3) + a*b**2*p**2*x**4*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*
b**3) + a*b**2*p*x**4*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) + b**3*p**2*x**6*(a +
b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) + 3*b**3*p*x**6*(a + b*x**2)**p/(2*b**3*p**3 +
12*b**3*p**2 + 22*b**3*p + 12*b**3) + 2*b**3*x**6*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12
*b**3), True)) + 3*d*e**2*Piecewise((a**p*x**8/8, Eq(b, 0)), (6*a**3*log(-I*sqrt(a)*sqrt(1/b) + x)/(12*a**3*b*
*4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*x**6) + 6*a**3*log(I*sqrt(a)*sqrt(1/b) + x)/(12*a**3*b**4 +
36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*x**6) + 11*a**3/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**
4 + 12*b**7*x**6) + 18*a**2*b*x**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6
*x**4 + 12*b**7*x**6) + 18*a**2*b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b
**6*x**4 + 12*b**7*x**6) + 27*a**2*b*x**2/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*x**6) +
18*a*b**2*x**4*log(-I*sqrt(a)*sqrt(1/b) + x)/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*x**
6) + 18*a*b**2*x**4*log(I*sqrt(a)*sqrt(1/b) + x)/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*
x**6) + 18*a*b**2*x**4/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*x**6) + 6*b**3*x**6*log(-I
*sqrt(a)*sqrt(1/b) + x)/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*x**6) + 6*b**3*x**6*log(I
*sqrt(a)*sqrt(1/b) + x)/(12*a**3*b**4 + 36*a**2*b**5*x**2 + 36*a*b**6*x**4 + 12*b**7*x**6), Eq(p, -4)), (-6*a*
*3*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) - 6*a**3*log(I*sqrt(a)*sqrt(1/b)
+ x)/(4*a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) - 9*a**3/(4*a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) - 12*a**
2*b*x**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) - 12*a**2*b*x**2*log(I*sqrt
(a)*sqrt(1/b) + x)/(4*a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) - 12*a**2*b*x**2/(4*a**2*b**4 + 8*a*b**5*x**2 +
4*b**6*x**4) - 6*a*b**2*x**4*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) - 6*a*
b**2*x**4*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) + 2*b**3*x**6/(4*a**2*b**4
+ 8*a*b**5*x**2 + 4*b**6*x**4), Eq(p, -3)), (6*a**3*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a*b**4 + 4*b**5*x**2) + 6
*a**3*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a*b**4 + 4*b**5*x**2) + 6*a**3/(4*a*b**4 + 4*b**5*x**2) + 6*a**2*b*x**2*
log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a*b**4 + 4*b**5*x**2) + 6*a**2*b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a*b**4
+ 4*b**5*x**2) - 3*a*b**2*x**4/(4*a*b**4 + 4*b**5*x**2) + b**3*x**6/(4*a*b**4 + 4*b**5*x**2), Eq(p, -2)), (-a*
*3*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*b**4) - a**3*log(I*sqrt(a)*sqrt(1/b) + x)/(2*b**4) + a**2*x**2/(2*b**3) -
a*x**4/(4*b**2) + x**6/(6*b), Eq(p, -1)), (-6*a**4*(a + b*x**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2
+ 100*b**4*p + 48*b**4) + 6*a**3*b*p*x**2*(a + b*x**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**
4*p + 48*b**4) - 3*a**2*b**2*p**2*x**4*(a + b*x**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p
+ 48*b**4) - 3*a**2*b**2*p*x**4*(a + b*x**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p + 48*
b**4) + a*b**3*p**3*x**6*(a + b*x**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p + 48*b**4) +
3*a*b**3*p**2*x**6*(a + b*x**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p + 48*b**4) + 2*a*b*
*3*p*x**6*(a + b*x**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p + 48*b**4) + b**4*p**3*x**8*
(a + b*x**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p + 48*b**4) + 6*b**4*p**2*x**8*(a + b*x
**2)**p/(2*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p + 48*b**4) + 11*b**4*p*x**8*(a + b*x**2)**p/(2
*b**4*p**4 + 20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p + 48*b**4) + 6*b**4*x**8*(a + b*x**2)**p/(2*b**4*p**4 +
20*b**4*p**3 + 70*b**4*p**2 + 100*b**4*p + 48*b**4), True))
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