3.23.82 \(\int (d+e x)^{5/2} (a+b x+c x^2)^3 \, dx\) [2282]

Optimal. Leaf size=286 \[ \frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{9/2}}{3 e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{15/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{17/2}}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \]

[Out]

2/7*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(7/2)/e^7-2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(9/2)/e^7+6/11*(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)^(11/2)/e^7-2/13*(-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)^(13/2)/e^7+2/5*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(15/2)/e^7-6/17*c^
2*(-b*e+2*c*d)*(e*x+d)^(17/2)/e^7+2/19*c^3*(e*x+d)^(19/2)/e^7

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \begin {gather*} \frac {2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{13/2} (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 )}{13 e^7}+\frac {6 (d+e x)^{11/2} \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 )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac {6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(
9/2))/(3*e^7) + (6*(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)^(11/2))/(11*e^7
) - (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)^(13/2))/(13*e^7) + (2*c*(5*c^2*d
^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(15/2))/(5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7)
+ (2*c^3*(d + e*x)^(19/2))/(19*e^7)

Rule 712

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] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{e^6}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{e^6}+\frac {3 \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)^{9/2}}{e^6}+\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)^{11/2}}{e^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{15/2}}{e^6}+\frac {c^3 (d+e x)^{17/2}}{e^6}\right ) \, dx\\ &=\frac {2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{7/2}}{7 e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{9/2}}{3 e^7}+\frac {6 \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)^{11/2}}{11 e^7}-\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)^{13/2}}{13 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{15/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{17/2}}{17 e^7}+\frac {2 c^3 (d+e x)^{19/2}}{19 e^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.31, size = 397, normalized size = 1.39 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (5 c^3 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )+1615 e^3 \left (429 a^3 e^3+143 a^2 b e^2 (-2 d+7 e x)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )+323 c e^2 \left (65 a^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 a b e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-19 c^2 e \left (-17 a e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+5 b \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{4849845 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(7/2)*(5*c^3*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4
- 36036*d*e^5*x^5 + 51051*e^6*x^6) + 1615*e^3*(429*a^3*e^3 + 143*a^2*b*e^2*(-2*d + 7*e*x) + 13*a*b^2*e*(8*d^2
- 28*d*e*x + 63*e^2*x^2) + b^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3)) + 323*c*e^2*(65*a^2*e^2*(
8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*a*b*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^2*(128*d^4
- 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)) - 19*c^2*e*(-17*a*e*(128*d^4 - 448*d^3*e*x
+ 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 5*b*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2 - 3696*d^2
*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5))))/(4849845*e^7)

________________________________________________________________________________________

Maple [A]
time = 0.73, size = 357, normalized size = 1.25 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2/e^7*(1/19*c^3*(e*x+d)^(19/2)+3/17*(b*e-2*c*d)*c^2*(e*x+d)^(17/2)+1/15*((a*e^2-b*d*e+c*d^2)*c^2+2*(b*e-2*c*d)
^2*c+c*(2*(a*e^2-b*d*e+c*d^2)*c+(b*e-2*c*d)^2))*(e*x+d)^(15/2)+1/13*(4*(a*e^2-b*d*e+c*d^2)*c*(b*e-2*c*d)+(b*e-
2*c*d)*(2*(a*e^2-b*d*e+c*d^2)*c+(b*e-2*c*d)^2))*(e*x+d)^(13/2)+1/11*((a*e^2-b*d*e+c*d^2)*(2*(a*e^2-b*d*e+c*d^2
)*c+(b*e-2*c*d)^2)+2*(b*e-2*c*d)^2*(a*e^2-b*d*e+c*d^2)+c*(a*e^2-b*d*e+c*d^2)^2)*(e*x+d)^(11/2)+1/3*(a*e^2-b*d*
e+c*d^2)^2*(b*e-2*c*d)*(e*x+d)^(9/2)+1/7*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(7/2))

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 426, normalized size = 1.49 \begin {gather*} \frac {2}{4849845} \, {\left (255255 \, {\left (x e + d\right )}^{\frac {19}{2}} c^{3} - 855855 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (x e + d\right )}^{\frac {17}{2}} + 969969 \, {\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 {15}{2}} - 373065 \, {\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 {13}{2}} + 1322685 \, {\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 {11}{2}} - 1616615 \, {\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 {9}{2}} + 692835 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} - 3 \, a^{2} b d e^{5} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + a^{3} e^{6} + 3 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

2/4849845*(255255*(x*e + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(x*e + d)^(17/2) + 969969*(5*c^3*d^2 - 5*b
*c^2*d*e + b^2*c*e^2 + a*c^2*e^2)*(x*e + d)^(15/2) - 373065*(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)^(13/2) + 1322685*(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)^(11/2) - 1616615*(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)^(9/2) + 692835*(c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c*e^2 + a*c^2*e^2)*d^4 - 3*a^2*b*d*e^5 - (b^3*e
^3 + 6*a*b*c*e^3)*d^3 + a^3*e^6 + 3*(a*b^2*e^4 + a^2*c*e^4)*d^2)*(x*e + d)^(7/2))*e^(-7)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (264) = 528\).
time = 3.60, size = 692, normalized size = 2.42 \begin {gather*} \frac {2}{4849845} \, {\left (5120 \, c^{3} d^{9} + {\left (255255 \, c^{3} x^{9} + 855855 \, b c^{2} x^{8} + 969969 \, {\left (b^{2} c + a c^{2}\right )} x^{7} + 1616615 \, a^{2} b x^{4} + 373065 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 692835 \, a^{3} x^{3} + 1322685 \, {\left (a b^{2} + a^{2} c\right )} x^{5}\right )} e^{9} + {\left (585585 \, c^{3} d x^{8} + 1996995 \, b c^{2} d x^{7} + 2313003 \, {\left (b^{2} c + a c^{2}\right )} d x^{6} + 4387955 \, a^{2} b d x^{3} + 915705 \, {\left (b^{3} + 6 \, a b c\right )} d x^{5} + 2078505 \, a^{3} d x^{2} + 3380195 \, {\left (a b^{2} + a^{2} c\right )} d x^{4}\right )} e^{8} + {\left (345345 \, c^{3} d^{2} x^{7} + 1206975 \, b c^{2} d^{2} x^{6} + 1444779 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{5} + 3464175 \, a^{2} b d^{2} x^{2} + 599165 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x^{4} + 2078505 \, a^{3} d^{2} x + 2372435 \, {\left (a b^{2} + a^{2} c\right )} d^{2} x^{3}\right )} e^{7} + 5 \, {\left (231 \, c^{3} d^{3} x^{6} + 1197 \, b c^{2} d^{3} x^{5} + 2261 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x^{4} + 46189 \, a^{2} b d^{3} x + 1615 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} x^{3} + 138567 \, a^{3} d^{3} + 12597 \, {\left (a b^{2} + a^{2} c\right )} d^{3} x^{2}\right )} e^{6} - 10 \, {\left (126 \, c^{3} d^{4} x^{5} + 665 \, b c^{2} d^{4} x^{4} + 1292 \, {\left (b^{2} c + a c^{2}\right )} d^{4} x^{3} + 46189 \, a^{2} b d^{4} + 969 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} x^{2} + 8398 \, {\left (a b^{2} + a^{2} c\right )} d^{4} x\right )} e^{5} + 8 \, {\left (175 \, c^{3} d^{5} x^{4} + 950 \, b c^{2} d^{5} x^{3} + 1938 \, {\left (b^{2} c + a c^{2}\right )} d^{5} x^{2} + 1615 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} x + 20995 \, {\left (a b^{2} + a^{2} c\right )} d^{5}\right )} e^{4} - 16 \, {\left (100 \, c^{3} d^{6} x^{3} + 570 \, b c^{2} d^{6} x^{2} + 1292 \, {\left (b^{2} c + a c^{2}\right )} d^{6} x + 1615 \, {\left (b^{3} + 6 \, a b c\right )} d^{6}\right )} e^{3} + 128 \, {\left (15 \, c^{3} d^{7} x^{2} + 95 \, b c^{2} d^{7} x + 323 \, {\left (b^{2} c + a c^{2}\right )} d^{7}\right )} e^{2} - 1280 \, {\left (2 \, c^{3} d^{8} x + 19 \, b c^{2} d^{8}\right )} e\right )} \sqrt {x e + d} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

2/4849845*(5120*c^3*d^9 + (255255*c^3*x^9 + 855855*b*c^2*x^8 + 969969*(b^2*c + a*c^2)*x^7 + 1616615*a^2*b*x^4
+ 373065*(b^3 + 6*a*b*c)*x^6 + 692835*a^3*x^3 + 1322685*(a*b^2 + a^2*c)*x^5)*e^9 + (585585*c^3*d*x^8 + 1996995
*b*c^2*d*x^7 + 2313003*(b^2*c + a*c^2)*d*x^6 + 4387955*a^2*b*d*x^3 + 915705*(b^3 + 6*a*b*c)*d*x^5 + 2078505*a^
3*d*x^2 + 3380195*(a*b^2 + a^2*c)*d*x^4)*e^8 + (345345*c^3*d^2*x^7 + 1206975*b*c^2*d^2*x^6 + 1444779*(b^2*c +
a*c^2)*d^2*x^5 + 3464175*a^2*b*d^2*x^2 + 599165*(b^3 + 6*a*b*c)*d^2*x^4 + 2078505*a^3*d^2*x + 2372435*(a*b^2 +
a^2*c)*d^2*x^3)*e^7 + 5*(231*c^3*d^3*x^6 + 1197*b*c^2*d^3*x^5 + 2261*(b^2*c + a*c^2)*d^3*x^4 + 46189*a^2*b*d^
3*x + 1615*(b^3 + 6*a*b*c)*d^3*x^3 + 138567*a^3*d^3 + 12597*(a*b^2 + a^2*c)*d^3*x^2)*e^6 - 10*(126*c^3*d^4*x^5
+ 665*b*c^2*d^4*x^4 + 1292*(b^2*c + a*c^2)*d^4*x^3 + 46189*a^2*b*d^4 + 969*(b^3 + 6*a*b*c)*d^4*x^2 + 8398*(a*
b^2 + a^2*c)*d^4*x)*e^5 + 8*(175*c^3*d^5*x^4 + 950*b*c^2*d^5*x^3 + 1938*(b^2*c + a*c^2)*d^5*x^2 + 1615*(b^3 +
6*a*b*c)*d^5*x + 20995*(a*b^2 + a^2*c)*d^5)*e^4 - 16*(100*c^3*d^6*x^3 + 570*b*c^2*d^6*x^2 + 1292*(b^2*c + a*c^
2)*d^6*x + 1615*(b^3 + 6*a*b*c)*d^6)*e^3 + 128*(15*c^3*d^7*x^2 + 95*b*c^2*d^7*x + 323*(b^2*c + a*c^2)*d^7)*e^2
- 1280*(2*c^3*d^8*x + 19*b*c^2*d^8)*e)*sqrt(x*e + d)*e^(-7)

________________________________________________________________________________________

Sympy [A]
time = 45.48, size = 2363, normalized size = 8.26 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(c*x**2+b*x+a)**3,x)

[Out]

a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**3*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
6*a**2*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*a**2*b*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(
d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*a**2*b*(-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**2 + 6*a**2*c*d**2*(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**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**3 + 6*a**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**3 + 6*a*b**2*d**2*(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**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**3 + 6*a*b**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**3
+ 12*a*b*c*d**2*(-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 + 24*a*b*c*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**4 + 12*a*b*c*(-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**4 + 6*a*c**2*d**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**5 + 12*a*c**2*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
**5 + 6*a*c**2*(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**5 + 2*b**
3*d**2*(-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 + 4*b**3*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**4 + 2*b**3*(-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**4 + 6*b**2
*c*d**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**5 + 12*b**2*c*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**5 + 6*b**2*
c*(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**5 + 6*b*c**2*d**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**6 + 12*b*c**2*d*(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 + 6*b*c**2*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3
*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)
/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**6 + 2*c**3*d**2*(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**7 + 4*c**3*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(
5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*
x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 2*c**3*(d**8*(d + e*x)**(3/2)/3 - 8*d*
*7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 -
56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)
/e**7

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3066 vs. \(2 (264) = 528\).
time = 1.56, size = 3066, normalized size = 10.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

2/4849845*(4849845*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d^3*e^(-1) + 969969*(3*(x*e + d)^(5/2) - 10*(x*
e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d^3*e^(-2) + 969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 1
5*sqrt(x*e + d)*d^2)*a^2*c*d^3*e^(-2) + 138567*(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^3*e^(-3) + 831402*(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^3*e^(-3) + 46189*(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^3*e^(-4) + 46189*(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^3*e^(-4) + 20995*(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^3*e^(-5) + 1615*(231*(x*e + d)^(13/2) - 16
38*(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^3*e^(-6) + 2909907*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)
*d + 15*sqrt(x*e + d)*d^2)*a^2*b*d^2*e^(-1) + 1247103*(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*d^2*e^(-2) + 1247103*(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*d^2*e^(-2) + 46189*(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*d^2*e^(-3) + 277134*(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*d^2*e^(-3) + 62985*(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*d^2*e^(-4) + 62985*(63*(x*e + d)^(1
1/2) - 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*d^2*e^(-4) + 14535*(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*d^2*e^(-5) + 2261*(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*d^2*e^(-6) + 4849845*sqrt(x*e + d)*a^3*d^3 + 4849845*((x*e + d)^(3/2) - 3*sq
rt(x*e + d)*d)*a^3*d^2 + 1247103*(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*b*d*e^(-1) + 138567*(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^2*d*e^(-2) + 138567*(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^2*c*d*e^(-2) + 20995*(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^3*d*e^(-3) + 125970*(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*b*c*d*e^(
-3) + 14535*(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^2*c*d*e^(-4) + 14535*(23
1*(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^2*d*e^(-4) + 6783*(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^2*d*e^(-5) + 133*(64
35*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 8
50850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d
^7 + 109395*sqrt(x*e + d)*d^8)*c^3*d*e^(-6) + 969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e +
d)*d^2)*a^3*d + 46189*(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^2*b*e^(-1) + 20995*(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*b^2*e^(-2) +
20995*(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^2*c*e^(-2) + 1615*(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)...

________________________________________________________________________________________

Mupad [B]
time = 0.10, size = 297, normalized size = 1.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{11\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {{\left (d+e\,x\right )}^{15/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{15\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{7\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{13\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x)

[Out]

((d + e*x)^(11/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 36*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2
- 60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/(11*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d
+ e*x)^(17/2))/(17*e^7) + ((d + e*x)^(15/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(15*e^7)
+ (2*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(7*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(13/2)*(b^2*e^2 + 10*c^2*
d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(13*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(3*e^7)

________________________________________________________________________________________