3.11.81 \(\int (b d+2 c d x)^{3/2} (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=88 \[ -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{72 c^3 d^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{80 c^3 d}+\frac {(b d+2 c d x)^{13/2}}{208 c^3 d^5} \]

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Rubi [A]  time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {683} \begin {gather*} -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{72 c^3 d^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{80 c^3 d}+\frac {(b d+2 c d x)^{13/2}}{208 c^3 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2,x]

[Out]

((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(5/2))/(80*c^3*d) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2))/(72*c^3*d^3) + (b*d
+ 2*c*d*x)^(13/2)/(208*c^3*d^5)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {align*} \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2 (b d+2 c d x)^{3/2}}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^{7/2}}{8 c^2 d^2}+\frac {(b d+2 c d x)^{11/2}}{16 c^2 d^4}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{80 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{72 c^3 d^3}+\frac {(b d+2 c d x)^{13/2}}{208 c^3 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 63, normalized size = 0.72 \begin {gather*} \frac {\left (-130 \left (b^2-4 a c\right ) (b+2 c x)^2+117 \left (b^2-4 a c\right )^2+45 (b+2 c x)^4\right ) (d (b+2 c x))^{5/2}}{9360 c^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2,x]

[Out]

((d*(b + 2*c*x))^(5/2)*(117*(b^2 - 4*a*c)^2 - 130*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 45*(b + 2*c*x)^4))/(9360*c^3*d
)

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IntegrateAlgebraic [A]  time = 0.10, size = 96, normalized size = 1.09 \begin {gather*} \frac {\left (117 a^2 c^2-26 a b^2 c+130 a b c^2 x+130 a c^3 x^2+2 b^4-10 b^3 c x+35 b^2 c^2 x^2+90 b c^3 x^3+45 c^4 x^4\right ) (b d+2 c d x)^{5/2}}{585 c^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2,x]

[Out]

((b*d + 2*c*d*x)^(5/2)*(2*b^4 - 26*a*b^2*c + 117*a^2*c^2 - 10*b^3*c*x + 130*a*b*c^2*x + 35*b^2*c^2*x^2 + 130*a
*c^3*x^2 + 90*b*c^3*x^3 + 45*c^4*x^4))/(585*c^3*d)

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fricas [B]  time = 0.39, size = 164, normalized size = 1.86 \begin {gather*} \frac {{\left (180 \, c^{6} d x^{6} + 540 \, b c^{5} d x^{5} + 5 \, {\left (109 \, b^{2} c^{4} + 104 \, a c^{5}\right )} d x^{4} + 10 \, {\left (19 \, b^{3} c^{3} + 104 \, a b c^{4}\right )} d x^{3} + 3 \, {\left (b^{4} c^{2} + 182 \, a b^{2} c^{3} + 156 \, a^{2} c^{4}\right )} d x^{2} - 2 \, {\left (b^{5} c - 13 \, a b^{3} c^{2} - 234 \, a^{2} b c^{3}\right )} d x + {\left (2 \, b^{6} - 26 \, a b^{4} c + 117 \, a^{2} b^{2} c^{2}\right )} d\right )} \sqrt {2 \, c d x + b d}}{585 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/585*(180*c^6*d*x^6 + 540*b*c^5*d*x^5 + 5*(109*b^2*c^4 + 104*a*c^5)*d*x^4 + 10*(19*b^3*c^3 + 104*a*b*c^4)*d*x
^3 + 3*(b^4*c^2 + 182*a*b^2*c^3 + 156*a^2*c^4)*d*x^2 - 2*(b^5*c - 13*a*b^3*c^2 - 234*a^2*b*c^3)*d*x + (2*b^6 -
26*a*b^4*c + 117*a^2*b^2*c^2)*d)*sqrt(2*c*d*x + b*d)/c^3

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giac [B]  time = 0.20, size = 867, normalized size = 9.85 \begin {gather*} \frac {720720 \, \sqrt {2 \, c d x + b d} a^{2} b^{2} d - 480480 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a^{2} b - \frac {240240 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a b^{3}}{c} + \frac {48048 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a^{2}}{d} + \frac {12012 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b^{4}}{c^{2} d} + \frac {120120 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a b^{2}}{c d} - \frac {15444 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b^{3}}{c^{2} d^{2}} - \frac {41184 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} a b}{c d^{2}} + \frac {1859 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} b^{2}}{c^{2} d^{3}} + \frac {1144 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} a}{c d^{3}} - \frac {390 \, {\left (693 \, \sqrt {2 \, c d x + b d} b^{5} d^{5} - 1155 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} d^{4} + 1386 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{3} d^{3} - 990 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{2} d^{2} + 385 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b d - 63 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}}\right )} b}{c^{2} d^{4}} + \frac {15 \, {\left (3003 \, \sqrt {2 \, c d x + b d} b^{6} d^{6} - 6006 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{5} d^{5} + 9009 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{4} d^{4} - 8580 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{3} d^{3} + 5005 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b^{2} d^{2} - 1638 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} b d + 231 \, {\left (2 \, c d x + b d\right )}^{\frac {13}{2}}\right )}}{c^{2} d^{5}}}{720720 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

1/720720*(720720*sqrt(2*c*d*x + b*d)*a^2*b^2*d - 480480*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*a^
2*b - 240240*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*a*b^3/c + 48048*(15*sqrt(2*c*d*x + b*d)*b^2*d
^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*a^2/d + 12012*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 1
0*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*b^4/(c^2*d) + 120120*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 -
10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*a*b^2/(c*d) - 15444*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 -
35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*b^3/(c^2*d^2) - 411
84*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*
d*x + b*d)^(7/2))*a*b/(c*d^2) + 1859*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 37
8*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))*b^2/(c^2*d^3) + 11
44*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^2 -
180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))*a/(c*d^3) - 390*(693*sqrt(2*c*d*x + b*d)*b^5*d^5 - 1
155*(2*c*d*x + b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5/2)*b^3*d^3 - 990*(2*c*d*x + b*d)^(7/2)*b^2*d^2 + 3
85*(2*c*d*x + b*d)^(9/2)*b*d - 63*(2*c*d*x + b*d)^(11/2))*b/(c^2*d^4) + 15*(3003*sqrt(2*c*d*x + b*d)*b^6*d^6 -
6006*(2*c*d*x + b*d)^(3/2)*b^5*d^5 + 9009*(2*c*d*x + b*d)^(5/2)*b^4*d^4 - 8580*(2*c*d*x + b*d)^(7/2)*b^3*d^3
+ 5005*(2*c*d*x + b*d)^(9/2)*b^2*d^2 - 1638*(2*c*d*x + b*d)^(11/2)*b*d + 231*(2*c*d*x + b*d)^(13/2))/(c^2*d^5)
)/c

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maple [A]  time = 0.05, size = 96, normalized size = 1.09 \begin {gather*} \frac {\left (2 c x +b \right ) \left (45 c^{4} x^{4}+90 b \,c^{3} x^{3}+130 a \,c^{3} x^{2}+35 x^{2} b^{2} c^{2}+130 a b \,c^{2} x -10 x \,b^{3} c +117 a^{2} c^{2}-26 a \,b^{2} c +2 b^{4}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{585 c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^2,x)

[Out]

1/585*(2*c*x+b)*(45*c^4*x^4+90*b*c^3*x^3+130*a*c^3*x^2+35*b^2*c^2*x^2+130*a*b*c^2*x-10*b^3*c*x+117*a^2*c^2-26*
a*b^2*c+2*b^4)*(2*c*d*x+b*d)^(3/2)/c^3

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maxima [A]  time = 1.46, size = 81, normalized size = 0.92 \begin {gather*} -\frac {130 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 117 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} d^{4} - 45 \, {\left (2 \, c d x + b d\right )}^{\frac {13}{2}}}{9360 \, c^{3} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

-1/9360*(130*(2*c*d*x + b*d)^(9/2)*(b^2 - 4*a*c)*d^2 - 117*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)^(5/2
)*d^4 - 45*(2*c*d*x + b*d)^(13/2))/(c^3*d^5)

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mupad [B]  time = 0.52, size = 99, normalized size = 1.12 \begin {gather*} \frac {{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (45\,{\left (b\,d+2\,c\,d\,x\right )}^4+117\,b^4\,d^4-130\,b^2\,d^2\,{\left (b\,d+2\,c\,d\,x\right )}^2+1872\,a^2\,c^2\,d^4+520\,a\,c\,d^2\,{\left (b\,d+2\,c\,d\,x\right )}^2-936\,a\,b^2\,c\,d^4\right )}{9360\,c^3\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2,x)

[Out]

((b*d + 2*c*d*x)^(5/2)*(45*(b*d + 2*c*d*x)^4 + 117*b^4*d^4 - 130*b^2*d^2*(b*d + 2*c*d*x)^2 + 1872*a^2*c^2*d^4
+ 520*a*c*d^2*(b*d + 2*c*d*x)^2 - 936*a*b^2*c*d^4))/(9360*c^3*d^5)

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sympy [A]  time = 19.76, size = 695, normalized size = 7.90 \begin {gather*} a^{2} b d \left (\begin {cases} x \sqrt {b d} & \text {for}\: c = 0 \\0 & \text {for}\: d = 0 \\\frac {\left (b d + 2 c d x\right )^{\frac {3}{2}}}{3 c d} & \text {otherwise} \end {cases}\right ) + \frac {a^{2} \left (- \frac {b d \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} + \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{c d} + \frac {a b^{2} \left (- \frac {b d \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} + \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{c^{2} d} + \frac {3 a b \left (\frac {b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} - \frac {2 b d \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{7}\right )}{2 c^{2} d^{2}} + \frac {a \left (- \frac {b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} + \frac {3 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} - \frac {3 b d \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} + \frac {\left (b d + 2 c d x\right )^{\frac {9}{2}}}{9}\right )}{2 c^{2} d^{3}} + \frac {b^{3} \left (\frac {b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} - \frac {2 b d \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{7}\right )}{4 c^{3} d^{2}} + \frac {b^{2} \left (- \frac {b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} + \frac {3 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} - \frac {3 b d \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} + \frac {\left (b d + 2 c d x\right )^{\frac {9}{2}}}{9}\right )}{2 c^{3} d^{3}} + \frac {5 b \left (\frac {b^{4} d^{4} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} - \frac {4 b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} + \frac {6 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} - \frac {4 b d \left (b d + 2 c d x\right )^{\frac {9}{2}}}{9} + \frac {\left (b d + 2 c d x\right )^{\frac {11}{2}}}{11}\right )}{16 c^{3} d^{4}} + \frac {- \frac {b^{5} d^{5} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} + b^{4} d^{4} \left (b d + 2 c d x\right )^{\frac {5}{2}} - \frac {10 b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} + \frac {10 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {9}{2}}}{9} - \frac {5 b d \left (b d + 2 c d x\right )^{\frac {11}{2}}}{11} + \frac {\left (b d + 2 c d x\right )^{\frac {13}{2}}}{13}}{16 c^{3} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(3/2)*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b*d*Piecewise((x*sqrt(b*d), Eq(c, 0)), (0, Eq(d, 0)), ((b*d + 2*c*d*x)**(3/2)/(3*c*d), True)) + a**2*(-b*
d*(b*d + 2*c*d*x)**(3/2)/3 + (b*d + 2*c*d*x)**(5/2)/5)/(c*d) + a*b**2*(-b*d*(b*d + 2*c*d*x)**(3/2)/3 + (b*d +
2*c*d*x)**(5/2)/5)/(c**2*d) + 3*a*b*(b**2*d**2*(b*d + 2*c*d*x)**(3/2)/3 - 2*b*d*(b*d + 2*c*d*x)**(5/2)/5 + (b*
d + 2*c*d*x)**(7/2)/7)/(2*c**2*d**2) + a*(-b**3*d**3*(b*d + 2*c*d*x)**(3/2)/3 + 3*b**2*d**2*(b*d + 2*c*d*x)**(
5/2)/5 - 3*b*d*(b*d + 2*c*d*x)**(7/2)/7 + (b*d + 2*c*d*x)**(9/2)/9)/(2*c**2*d**3) + b**3*(b**2*d**2*(b*d + 2*c
*d*x)**(3/2)/3 - 2*b*d*(b*d + 2*c*d*x)**(5/2)/5 + (b*d + 2*c*d*x)**(7/2)/7)/(4*c**3*d**2) + b**2*(-b**3*d**3*(
b*d + 2*c*d*x)**(3/2)/3 + 3*b**2*d**2*(b*d + 2*c*d*x)**(5/2)/5 - 3*b*d*(b*d + 2*c*d*x)**(7/2)/7 + (b*d + 2*c*d
*x)**(9/2)/9)/(2*c**3*d**3) + 5*b*(b**4*d**4*(b*d + 2*c*d*x)**(3/2)/3 - 4*b**3*d**3*(b*d + 2*c*d*x)**(5/2)/5 +
6*b**2*d**2*(b*d + 2*c*d*x)**(7/2)/7 - 4*b*d*(b*d + 2*c*d*x)**(9/2)/9 + (b*d + 2*c*d*x)**(11/2)/11)/(16*c**3*
d**4) + (-b**5*d**5*(b*d + 2*c*d*x)**(3/2)/3 + b**4*d**4*(b*d + 2*c*d*x)**(5/2) - 10*b**3*d**3*(b*d + 2*c*d*x)
**(7/2)/7 + 10*b**2*d**2*(b*d + 2*c*d*x)**(9/2)/9 - 5*b*d*(b*d + 2*c*d*x)**(11/2)/11 + (b*d + 2*c*d*x)**(13/2)
/13)/(16*c**3*d**5)

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