3.11.68 \(\int (A+B x) (d+e x)^{7/2} (b x+c x^2) \, dx\)

Optimal. Leaf size=126 \[ -\frac {2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac {2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac {2 B c (d+e x)^{15/2}}{15 e^4} \]

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Rubi [A]  time = 0.09, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac {2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac {2 B c (d+e x)^{15/2}}{15 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2),x]

[Out]

(-2*d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^(9/2))/(9*e^4) + (2*(B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))*(d + e*x
)^(11/2))/(11*e^4) - (2*(3*B*c*d - b*B*e - A*c*e)*(d + e*x)^(13/2))/(13*e^4) + (2*B*c*(d + e*x)^(15/2))/(15*e^
4)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e) (d+e x)^{7/2}}{e^3}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{9/2}}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^{11/2}}{e^3}+\frac {B c (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac {2 d (B d-A e) (c d-b e) (d+e x)^{9/2}}{9 e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{11/2}}{11 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{13/2}}{13 e^4}+\frac {2 B c (d+e x)^{15/2}}{15 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 113, normalized size = 0.90 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (5 A e \left (13 b e (9 e x-2 d)+c \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+B \left (5 b e \left (8 d^2-36 d e x+99 e^2 x^2\right )+c \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )\right )}{6435 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(9/2)*(5*A*e*(13*b*e*(-2*d + 9*e*x) + c*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) + B*(5*b*e*(8*d^2 - 36*d
*e*x + 99*e^2*x^2) + c*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3))))/(6435*e^4)

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IntegrateAlgebraic [A]  time = 0.09, size = 141, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (585 A b e^2 (d+e x)-715 A b d e^2+715 A c d^2 e-1170 A c d e (d+e x)+495 A c e (d+e x)^2+715 b B d^2 e-1170 b B d e (d+e x)+495 b B e (d+e x)^2-715 B c d^3+1755 B c d^2 (d+e x)-1485 B c d (d+e x)^2+429 B c (d+e x)^3\right )}{6435 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(9/2)*(-715*B*c*d^3 + 715*b*B*d^2*e + 715*A*c*d^2*e - 715*A*b*d*e^2 + 1755*B*c*d^2*(d + e*x) - 11
70*b*B*d*e*(d + e*x) - 1170*A*c*d*e*(d + e*x) + 585*A*b*e^2*(d + e*x) - 1485*B*c*d*(d + e*x)^2 + 495*b*B*e*(d
+ e*x)^2 + 495*A*c*e*(d + e*x)^2 + 429*B*c*(d + e*x)^3))/(6435*e^4)

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fricas [B]  time = 0.41, size = 271, normalized size = 2.15 \begin {gather*} \frac {2 \, {\left (429 \, B c e^{7} x^{7} - 16 \, B c d^{7} - 130 \, A b d^{5} e^{2} + 40 \, {\left (B b + A c\right )} d^{6} e + 33 \, {\left (46 \, B c d e^{6} + 15 \, {\left (B b + A c\right )} e^{7}\right )} x^{6} + 9 \, {\left (206 \, B c d^{2} e^{5} + 65 \, A b e^{7} + 200 \, {\left (B b + A c\right )} d e^{6}\right )} x^{5} + 10 \, {\left (80 \, B c d^{3} e^{4} + 221 \, A b d e^{6} + 229 \, {\left (B b + A c\right )} d^{2} e^{5}\right )} x^{4} + 5 \, {\left (B c d^{4} e^{3} + 598 \, A b d^{2} e^{5} + 212 \, {\left (B b + A c\right )} d^{3} e^{4}\right )} x^{3} - 3 \, {\left (2 \, B c d^{5} e^{2} - 520 \, A b d^{3} e^{4} - 5 \, {\left (B b + A c\right )} d^{4} e^{3}\right )} x^{2} + {\left (8 \, B c d^{6} e + 65 \, A b d^{4} e^{3} - 20 \, {\left (B b + A c\right )} d^{5} e^{2}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="fricas")

[Out]

2/6435*(429*B*c*e^7*x^7 - 16*B*c*d^7 - 130*A*b*d^5*e^2 + 40*(B*b + A*c)*d^6*e + 33*(46*B*c*d*e^6 + 15*(B*b + A
*c)*e^7)*x^6 + 9*(206*B*c*d^2*e^5 + 65*A*b*e^7 + 200*(B*b + A*c)*d*e^6)*x^5 + 10*(80*B*c*d^3*e^4 + 221*A*b*d*e
^6 + 229*(B*b + A*c)*d^2*e^5)*x^4 + 5*(B*c*d^4*e^3 + 598*A*b*d^2*e^5 + 212*(B*b + A*c)*d^3*e^4)*x^3 - 3*(2*B*c
*d^5*e^2 - 520*A*b*d^3*e^4 - 5*(B*b + A*c)*d^4*e^3)*x^2 + (8*B*c*d^6*e + 65*A*b*d^4*e^3 - 20*(B*b + A*c)*d^5*e
^2)*x)*sqrt(e*x + d)/e^4

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giac [B]  time = 0.24, size = 1383, normalized size = 10.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*b*d^4*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(
3/2)*d + 15*sqrt(x*e + d)*d^2)*B*b*d^4*e^(-2) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e +
d)*d^2)*A*c*d^4*e^(-2) + 1287*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*
e + d)*d^3)*B*c*d^4*e^(-3) + 12012*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b*d^3*e
^(-1) + 5148*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*b*d^
3*e^(-2) + 5148*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*c
*d^3*e^(-2) + 572*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*
d^3 + 315*sqrt(x*e + d)*d^4)*B*c*d^3*e^(-3) + 7722*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3
/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b*d^2*e^(-1) + 858*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e +
d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b*d^2*e^(-2) + 858*(35*(x*e + d)^(9/2) - 180
*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*c*d^2*e^(-2)
+ 390*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 115
5*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*c*d^2*e^(-3) + 572*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*
d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*b*d*e^(-1) + 260*(63*(x*e + d
)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d
^4 - 693*sqrt(x*e + d)*d^5)*B*b*d*e^(-2) + 260*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7
/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*c*d*e^(-2) + 60*(231*
(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e +
d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*c*d*e^(-3) + 65*(63*(x*e + d)^(11/2) - 38
5*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt
(x*e + d)*d^5)*A*b*e^(-1) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 85
80*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*b*e^(
-2) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3
+ 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*c*e^(-2) + 7*(429*(x*e + d)
^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(
7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*c*e^(-3))*e^(-1)

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maple [A]  time = 0.05, size = 121, normalized size = 0.96 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (-429 B c \,x^{3} e^{3}-495 A c \,e^{3} x^{2}-495 B b \,e^{3} x^{2}+198 B c d \,e^{2} x^{2}-585 A b \,e^{3} x +180 A c d \,e^{2} x +180 B b d \,e^{2} x -72 B c \,d^{2} e x +130 A b d \,e^{2}-40 A c \,d^{2} e -40 B b \,d^{2} e +16 B c \,d^{3}\right )}{6435 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x)

[Out]

-2/6435*(e*x+d)^(9/2)*(-429*B*c*e^3*x^3-495*A*c*e^3*x^2-495*B*b*e^3*x^2+198*B*c*d*e^2*x^2-585*A*b*e^3*x+180*A*
c*d*e^2*x+180*B*b*d*e^2*x-72*B*c*d^2*e*x+130*A*b*d*e^2-40*A*c*d^2*e-40*B*b*d^2*e+16*B*c*d^3)/e^4

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maxima [A]  time = 0.57, size = 112, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} B c - 495 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 585 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="maxima")

[Out]

2/6435*(429*(e*x + d)^(15/2)*B*c - 495*(3*B*c*d - (B*b + A*c)*e)*(e*x + d)^(13/2) + 585*(3*B*c*d^2 + A*b*e^2 -
2*(B*b + A*c)*d*e)*(e*x + d)^(11/2) - 715*(B*c*d^3 + A*b*d*e^2 - (B*b + A*c)*d^2*e)*(e*x + d)^(9/2))/e^4

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mupad [B]  time = 0.10, size = 111, normalized size = 0.88 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{11\,e^4}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{13\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)*(A + B*x)*(d + e*x)^(7/2),x)

[Out]

((d + e*x)^(11/2)*(2*A*b*e^2 + 6*B*c*d^2 - 4*A*c*d*e - 4*B*b*d*e))/(11*e^4) + ((d + e*x)^(13/2)*(2*A*c*e + 2*B
*b*e - 6*B*c*d))/(13*e^4) + (2*B*c*(d + e*x)^(15/2))/(15*e^4) - (2*d*(A*e - B*d)*(b*e - c*d)*(d + e*x)^(9/2))/
(9*e^4)

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sympy [A]  time = 9.27, size = 683, normalized size = 5.42 \begin {gather*} \begin {cases} - \frac {4 A b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 A b d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 A b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 A b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 A b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 A b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 A c d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 A c d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 A c d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 A c d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 A c d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 A c d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 A c e^{3} x^{6} \sqrt {d + e x}}{13} + \frac {16 B b d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 B b d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 B b d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 B b d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 B b d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 B b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 B b e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 B c d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 B c d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 B c d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 B c d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 B c d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 B c d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 B c d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 B c e^{3} x^{7} \sqrt {d + e x}}{15} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {A b x^{2}}{2} + \frac {A c x^{3}}{3} + \frac {B b x^{3}}{3} + \frac {B c x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(7/2)*(c*x**2+b*x),x)

[Out]

Piecewise((-4*A*b*d**5*sqrt(d + e*x)/(99*e**2) + 2*A*b*d**4*x*sqrt(d + e*x)/(99*e) + 16*A*b*d**3*x**2*sqrt(d +
e*x)/33 + 92*A*b*d**2*e*x**3*sqrt(d + e*x)/99 + 68*A*b*d*e**2*x**4*sqrt(d + e*x)/99 + 2*A*b*e**3*x**5*sqrt(d
+ e*x)/11 + 16*A*c*d**6*sqrt(d + e*x)/(1287*e**3) - 8*A*c*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*A*c*d**4*x**2*s
qrt(d + e*x)/(429*e) + 424*A*c*d**3*x**3*sqrt(d + e*x)/1287 + 916*A*c*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*A*c*
d*e**2*x**5*sqrt(d + e*x)/143 + 2*A*c*e**3*x**6*sqrt(d + e*x)/13 + 16*B*b*d**6*sqrt(d + e*x)/(1287*e**3) - 8*B
*b*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*B*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*B*b*d**3*x**3*sqrt(d + e*x)/
1287 + 916*B*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*B*b*d*e**2*x**5*sqrt(d + e*x)/143 + 2*B*b*e**3*x**6*sqrt(d
+ e*x)/13 - 32*B*c*d**7*sqrt(d + e*x)/(6435*e**4) + 16*B*c*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*B*c*d**5*x**2*
sqrt(d + e*x)/(2145*e**2) + 2*B*c*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*B*c*d**3*x**4*sqrt(d + e*x)/1287 + 41
2*B*c*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B*c*d*e**2*x**6*sqrt(d + e*x)/195 + 2*B*c*e**3*x**7*sqrt(d + e*x)/15,
Ne(e, 0)), (d**(7/2)*(A*b*x**2/2 + A*c*x**3/3 + B*b*x**3/3 + B*c*x**4/4), True))

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