3.17.2 \(\int (b+2 c x) (d+e x)^{3/2} (a+b x+c x^2)^2 \, dx\) [1602]

Optimal. Leaf size=252 \[ -\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{5 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{9/2}}{9 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^6}+\frac {4 c^3 (d+e x)^{15/2}}{15 e^6} \]

[Out]

-2/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(5/2)/e^6+4/7*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e
+5*b*d))*(e*x+d)^(7/2)/e^6-2/9*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(9/2)/e^6+8/11*c
*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(11/2)/e^6-10/13*c^2*(-b*e+2*c*d)*(e*x+d)^(13/2)/e^6+4/15*c^3*(e
*x+d)^(15/2)/e^6

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Rubi [A]
time = 0.08, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {785} \begin {gather*} \frac {8 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^6}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^6}+\frac {4 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {10 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^6}+\frac {4 c^3 (d+e x)^{15/2}}{15 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2))/(5*e^6) + (4*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 +
b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(7/2))/(7*e^6) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d
- 3*a*e))*(d + e*x)^(9/2))/(9*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(11/2))/(11*e^6
) - (10*c^2*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^6) + (4*c^3*(d + e*x)^(15/2))/(15*e^6)

Rule 785

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 (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{e^5}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{5/2}}{e^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{e^5}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{11/2}}{e^5}+\frac {2 c^3 (d+e x)^{13/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{5 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{9/2}}{9 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^6}+\frac {4 c^3 (d+e x)^{15/2}}{15 e^6}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 291, normalized size = 1.15 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (c^3 \left (-512 d^5+1280 d^4 e x-2240 d^3 e^2 x^2+3360 d^2 e^3 x^3-4620 d e^4 x^4+6006 e^5 x^5\right )+143 b e^3 \left (63 a^2 e^2+18 a b e (-2 d+5 e x)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-78 c e^2 \left (33 a^2 e^2 (2 d-5 e x)-11 a b e \left (8 d^2-20 d e x+35 e^2 x^2\right )+2 b^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )+3 c^2 e \left (52 a e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+5 b \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(c^3*(-512*d^5 + 1280*d^4*e*x - 2240*d^3*e^2*x^2 + 3360*d^2*e^3*x^3 - 4620*d*e^4*x^4 + 6006
*e^5*x^5) + 143*b*e^3*(63*a^2*e^2 + 18*a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 78*c*e^2*
(33*a^2*e^2*(2*d - 5*e*x) - 11*a*b*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 2*b^2*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x
^2 - 105*e^3*x^3)) + 3*c^2*e*(52*a*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 5*b*(128*d^4 - 320*
d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4))))/(45045*e^6)

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Maple [A]
time = 0.89, size = 265, normalized size = 1.05 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/e^6*(2/15*c^3*(e*x+d)^(15/2)+5/13*(b*e-2*c*d)*c^2*(e*x+d)^(13/2)+1/11*(2*(b*e-2*c*d)^2*c+2*c*((b*e-2*c*d)^2+
2*c*(a*e^2-b*d*e+c*d^2)))*(e*x+d)^(11/2)+1/9*((b*e-2*c*d)*((b*e-2*c*d)^2+2*c*(a*e^2-b*d*e+c*d^2))+4*c*(a*e^2-b
*d*e+c*d^2)*(b*e-2*c*d))*(e*x+d)^(9/2)+1/7*(2*(b*e-2*c*d)^2*(a*e^2-b*d*e+c*d^2)+2*c*(a*e^2-b*d*e+c*d^2)^2)*(e*
x+d)^(7/2)+1/5*(b*e-2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(5/2))

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Maxima [A]
time = 0.28, size = 323, normalized size = 1.28 \begin {gather*} \frac {2}{45045} \, {\left (6006 \, {\left (x e + d\right )}^{\frac {15}{2}} c^{3} - 17325 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {13}{2}} + 16380 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {11}{2}} - 5005 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e - b^{3} e^{3} - 6 \, a b c e^{3} + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 12870 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + a b^{2} e^{4} + a^{2} c e^{4} + 6 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{2} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 9009 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{3} - a^{2} b e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{2} + 2 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(6006*(x*e + d)^(15/2)*c^3 - 17325*(2*c^3*d - b*c^2*e)*(x*e + d)^(13/2) + 16380*(5*c^3*d^2 - 5*b*c^2*d
*e + b^2*c*e^2 + a*c^2*e^2)*(x*e + d)^(11/2) - 5005*(20*c^3*d^3 - 30*b*c^2*d^2*e - b^3*e^3 - 6*a*b*c*e^3 + 12*
(b^2*c*e^2 + a*c^2*e^2)*d)*(x*e + d)^(9/2) + 12870*(5*c^3*d^4 - 10*b*c^2*d^3*e + a*b^2*e^4 + a^2*c*e^4 + 6*(b^
2*c*e^2 + a*c^2*e^2)*d^2 - (b^3*e^3 + 6*a*b*c*e^3)*d)*(x*e + d)^(7/2) - 9009*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*(b
^2*c*e^2 + a*c^2*e^2)*d^3 - a^2*b*e^5 - (b^3*e^3 + 6*a*b*c*e^3)*d^2 + 2*(a*b^2*e^4 + a^2*c*e^4)*d)*(x*e + d)^(
5/2))*e^(-6)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (234) = 468\).
time = 1.98, size = 470, normalized size = 1.87 \begin {gather*} -\frac {2}{45045} \, {\left (512 \, c^{3} d^{7} - {\left (6006 \, c^{3} x^{7} + 17325 \, b c^{2} x^{6} + 16380 \, {\left (b^{2} c + a c^{2}\right )} x^{5} + 9009 \, a^{2} b x^{2} + 5005 \, {\left (b^{3} + 6 \, a b c\right )} x^{4} + 12870 \, {\left (a b^{2} + a^{2} c\right )} x^{3}\right )} e^{7} - 2 \, {\left (3696 \, c^{3} d x^{6} + 11025 \, b c^{2} d x^{5} + 10920 \, {\left (b^{2} c + a c^{2}\right )} d x^{4} + 9009 \, a^{2} b d x + 3575 \, {\left (b^{3} + 6 \, a b c\right )} d x^{3} + 10296 \, {\left (a b^{2} + a^{2} c\right )} d x^{2}\right )} e^{6} - 3 \, {\left (42 \, c^{3} d^{2} x^{5} + 175 \, b c^{2} d^{2} x^{4} + 260 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{3} + 3003 \, a^{2} b d^{2} + 143 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x^{2} + 858 \, {\left (a b^{2} + a^{2} c\right )} d^{2} x\right )} e^{5} + 4 \, {\left (35 \, c^{3} d^{3} x^{4} + 150 \, b c^{2} d^{3} x^{3} + 234 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x^{2} + 143 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} x + 1287 \, {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} e^{4} - 8 \, {\left (20 \, c^{3} d^{4} x^{3} + 90 \, b c^{2} d^{4} x^{2} + 156 \, {\left (b^{2} c + a c^{2}\right )} d^{4} x + 143 \, {\left (b^{3} + 6 \, a b c\right )} d^{4}\right )} e^{3} + 192 \, {\left (c^{3} d^{5} x^{2} + 5 \, b c^{2} d^{5} x + 13 \, {\left (b^{2} c + a c^{2}\right )} d^{5}\right )} e^{2} - 128 \, {\left (2 \, c^{3} d^{6} x + 15 \, b c^{2} d^{6}\right )} e\right )} \sqrt {x e + d} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45045*(512*c^3*d^7 - (6006*c^3*x^7 + 17325*b*c^2*x^6 + 16380*(b^2*c + a*c^2)*x^5 + 9009*a^2*b*x^2 + 5005*(b
^3 + 6*a*b*c)*x^4 + 12870*(a*b^2 + a^2*c)*x^3)*e^7 - 2*(3696*c^3*d*x^6 + 11025*b*c^2*d*x^5 + 10920*(b^2*c + a*
c^2)*d*x^4 + 9009*a^2*b*d*x + 3575*(b^3 + 6*a*b*c)*d*x^3 + 10296*(a*b^2 + a^2*c)*d*x^2)*e^6 - 3*(42*c^3*d^2*x^
5 + 175*b*c^2*d^2*x^4 + 260*(b^2*c + a*c^2)*d^2*x^3 + 3003*a^2*b*d^2 + 143*(b^3 + 6*a*b*c)*d^2*x^2 + 858*(a*b^
2 + a^2*c)*d^2*x)*e^5 + 4*(35*c^3*d^3*x^4 + 150*b*c^2*d^3*x^3 + 234*(b^2*c + a*c^2)*d^3*x^2 + 143*(b^3 + 6*a*b
*c)*d^3*x + 1287*(a*b^2 + a^2*c)*d^3)*e^4 - 8*(20*c^3*d^4*x^3 + 90*b*c^2*d^4*x^2 + 156*(b^2*c + a*c^2)*d^4*x +
143*(b^3 + 6*a*b*c)*d^4)*e^3 + 192*(c^3*d^5*x^2 + 5*b*c^2*d^5*x + 13*(b^2*c + a*c^2)*d^5)*e^2 - 128*(2*c^3*d^
6*x + 15*b*c^2*d^6)*e)*sqrt(x*e + d)*e^(-6)

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Sympy [A]
time = 20.11, size = 1093, normalized size = 4.34 \begin {gather*} a^{2} b d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{2} b \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {4 a^{2} c d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {4 a^{2} c \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {4 a b^{2} d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {4 a b^{2} \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {12 a b c d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {12 a b c \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {8 a c^{2} d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {8 a c^{2} \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} + \frac {2 b^{3} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 b^{3} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {8 b^{2} c d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {8 b^{2} c \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} + \frac {10 b c^{2} d \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {10 b c^{2} \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} + \frac {4 c^{3} d \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{6}} + \frac {4 c^{3} \left (\frac {d^{6} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {6 d^{5} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {15 d^{4} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {20 d^{3} \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {15 d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {6 d \left (d + e x\right )^{\frac {13}{2}}}{13} + \frac {\left (d + e x\right )^{\frac {15}{2}}}{15}\right )}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*b*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**2*b*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 4*a**2*c*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*a**2*c*(d**2*(d + e
*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 4*a*b**2*d*(-d*(d + e*x)**(3/2)/3 + (d + e*
x)**(5/2)/5)/e**2 + 4*a*b**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 12
*a*b*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*a*b*c*(-d**3*(d + e
*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 8*a*c**2*d*(-d*
*3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 8*a*c*
*2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 +
(d + e*x)**(11/2)/11)/e**4 + 2*b**3*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)
/e**3 + 2*b**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/
2)/9)/e**3 + 8*b**2*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d +
e*x)**(9/2)/9)/e**4 + 8*b**2*c*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/
7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 10*b*c**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e
*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 10*b*c**2*(-d
**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d
*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 4*c**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2
) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)
/e**6 + 4*c**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d
+ e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1648 vs. \(2 (234) = 468\).
time = 1.87, size = 1648, normalized size = 6.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b^2*d^2*e^(-1) + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*a^2*c*d^2*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^3*d^2*e^(-2)
+ 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b*c*d^2*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)*b^2*c*d^2*e^(-3) + 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^2*d^2*e^(-3) + 715*(3
5*(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^2*d^2*e^(-4) + 130*(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)*c^3*d^2*e^(-5) + 12012*(3*(x*e + d)^(5
/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d*e^(-1) + 12012*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3
/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d*e^(-1) + 2574*(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^3*d*e^(-2) + 15444*(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*c*d*e^(-2) + 1144*(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^2*c*d*e^(-3) + 1144*(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^2*d*e
^(-3) + 650*(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*c^2*d*e^(-4) + 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)*c^3*d*e^(-5) + 45045*sqrt(x*e + d)*a^2*b*d^2 + 30030*((x*e + d)^(3/2) - 3*sq
rt(x*e + d)*d)*a^2*b*d + 2574*(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^2*e^(-1) + 2574*(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^2*c*e^(-1) + 143*(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^3*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*b*c*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)*b^2*c*e^(-3) + 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^2*e^(-3) + 75*(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^2*e^(-4) + 14*(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)*c^3*e^(-5) + 3003*(3*(x*e + d)^(5/2) -
10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b)*e^(-1)

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Mupad [B]
time = 0.07, size = 267, normalized size = 1.06 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{7\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{11\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{9\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{5\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(7/2)*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 24*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 -
40*b*c^2*d^3*e - 24*a*b*c*d*e^3))/(7*e^6) + (4*c^3*(d + e*x)^(15/2))/(15*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d +
e*x)^(13/2))/(13*e^6) + ((d + e*x)^(11/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2*c*e^2 - 40*b*c^2*d*e))/(11*e^6) +
(2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(9*e^6) + (2*(b*e - 2*c*d)*
(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(5*e^6)

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