3.2282 \(\int (d+e x)^{5/2} (a+b x+c x^2)^3 \, dx\)
Optimal. Leaf size=286 \[ \frac{2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{13 e^7}+\frac{6 (d+e x)^{11/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7} \]
[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)
________________________________________________________________________________________
Rubi [A] time = 0.180941, antiderivative size = 286, normalized size of antiderivative = 1.,
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 =
{698} \[ \frac{2 c (d+e x)^{15/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{13 e^7}+\frac{6 (d+e x)^{11/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^3}{7 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7} \]
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 698
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 = 0.909399, size = 321, normalized size = 1.12 \[ \frac{2 \left ((d+e x)^{7/2} (a+x (b+c x))^3-\frac{2 (d+e x)^{9/2} \left (-646 c e^2 \left (65 a^2 e^2 (2 d-9 e x)-15 a b e \left (8 d^2-36 d e x+99 e^2 x^2\right )+2 b^2 \left (-72 d^2 e x+16 d^3+198 d e^2 x^2-429 e^3 x^3\right )\right )+1615 b e^3 \left (143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+19 c^2 e \left (68 a e \left (72 d^2 e x-16 d^3-198 d e^2 x^2+429 e^3 x^3\right )+5 b \left (1584 d^2 e^2 x^2-576 d^3 e x+128 d^4-3432 d e^3 x^3+6435 e^4 x^4\right )\right )-10 c^3 \left (3168 d^3 e^2 x^2-6864 d^2 e^3 x^3-1152 d^4 e x+256 d^5+12870 d e^4 x^4-21879 e^5 x^5\right )\right )}{692835 e^6}\right )}{7 e} \]
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)*(a + x*(b + c*x))^3 - (2*(d + e*x)^(9/2)*(-10*c^3*(256*d^5 - 1152*d^4*e*x + 3168*d^3*e^2*x
^2 - 6864*d^2*e^3*x^3 + 12870*d*e^4*x^4 - 21879*e^5*x^5) + 1615*b*e^3*(143*a^2*e^2 + 26*a*b*e*(-2*d + 9*e*x) +
b^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) - 646*c*e^2*(65*a^2*e^2*(2*d - 9*e*x) - 15*a*b*e*(8*d^2 - 36*d*e*x + 99*
e^2*x^2) + 2*b^2*(16*d^3 - 72*d^2*e*x + 198*d*e^2*x^2 - 429*e^3*x^3)) + 19*c^2*e*(68*a*e*(-16*d^3 + 72*d^2*e*x
- 198*d*e^2*x^2 + 429*e^3*x^3) + 5*b*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^
4))))/(692835*e^6)))/(7*e)
________________________________________________________________________________________
Maple [A] time = 0.046, size = 495, normalized size = 1.7 \begin{align*}{\frac{510510\,{c}^{3}{x}^{6}{e}^{6}+1711710\,b{c}^{2}{e}^{6}{x}^{5}-360360\,{c}^{3}d{e}^{5}{x}^{5}+1939938\,a{c}^{2}{e}^{6}{x}^{4}+1939938\,{b}^{2}c{e}^{6}{x}^{4}-1141140\,b{c}^{2}d{e}^{5}{x}^{4}+240240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+4476780\,abc{e}^{6}{x}^{3}-1193808\,a{c}^{2}d{e}^{5}{x}^{3}+746130\,{b}^{3}{e}^{6}{x}^{3}-1193808\,{b}^{2}cd{e}^{5}{x}^{3}+702240\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-147840\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+2645370\,{a}^{2}c{e}^{6}{x}^{2}+2645370\,a{b}^{2}{e}^{6}{x}^{2}-2441880\,abcd{e}^{5}{x}^{2}+651168\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-406980\,{b}^{3}d{e}^{5}{x}^{2}+651168\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-383040\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+80640\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+3233230\,{a}^{2}b{e}^{6}x-1175720\,{a}^{2}cd{e}^{5}x-1175720\,a{b}^{2}d{e}^{5}x+1085280\,abc{d}^{2}{e}^{4}x-289408\,a{c}^{2}{d}^{3}{e}^{3}x+180880\,{b}^{3}{d}^{2}{e}^{4}x-289408\,{b}^{2}c{d}^{3}{e}^{3}x+170240\,b{c}^{2}{d}^{4}{e}^{2}x-35840\,{c}^{3}{d}^{5}ex+1385670\,{a}^{3}{e}^{6}-923780\,{a}^{2}bd{e}^{5}+335920\,{a}^{2}c{d}^{2}{e}^{4}+335920\,a{b}^{2}{d}^{2}{e}^{4}-310080\,abc{d}^{3}{e}^{3}+82688\,a{c}^{2}{d}^{4}{e}^{2}-51680\,{b}^{3}{d}^{3}{e}^{3}+82688\,{b}^{2}c{d}^{4}{e}^{2}-48640\,b{c}^{2}{d}^{5}e+10240\,{c}^{3}{d}^{6}}{4849845\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
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)
[Out]
2/4849845*(e*x+d)^(7/2)*(255255*c^3*e^6*x^6+855855*b*c^2*e^6*x^5-180180*c^3*d*e^5*x^5+969969*a*c^2*e^6*x^4+969
969*b^2*c*e^6*x^4-570570*b*c^2*d*e^5*x^4+120120*c^3*d^2*e^4*x^4+2238390*a*b*c*e^6*x^3-596904*a*c^2*d*e^5*x^3+3
73065*b^3*e^6*x^3-596904*b^2*c*d*e^5*x^3+351120*b*c^2*d^2*e^4*x^3-73920*c^3*d^3*e^3*x^3+1322685*a^2*c*e^6*x^2+
1322685*a*b^2*e^6*x^2-1220940*a*b*c*d*e^5*x^2+325584*a*c^2*d^2*e^4*x^2-203490*b^3*d*e^5*x^2+325584*b^2*c*d^2*e
^4*x^2-191520*b*c^2*d^3*e^3*x^2+40320*c^3*d^4*e^2*x^2+1616615*a^2*b*e^6*x-587860*a^2*c*d*e^5*x-587860*a*b^2*d*
e^5*x+542640*a*b*c*d^2*e^4*x-144704*a*c^2*d^3*e^3*x+90440*b^3*d^2*e^4*x-144704*b^2*c*d^3*e^3*x+85120*b*c^2*d^4
*e^2*x-17920*c^3*d^5*e*x+692835*a^3*e^6-461890*a^2*b*d*e^5+167960*a^2*c*d^2*e^4+167960*a*b^2*d^2*e^4-155040*a*
b*c*d^3*e^3+41344*a*c^2*d^4*e^2-25840*b^3*d^3*e^3+41344*b^2*c*d^4*e^2-24320*b*c^2*d^5*e+5120*c^3*d^6)/e^7
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Maxima [A] time = 1.01909, size = 549, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (255255 \,{\left (e x + d\right )}^{\frac{19}{2}} c^{3} - 855855 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 969969 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 373065 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 1616615 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 692835 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{4849845 \, e^{7}} \end{align*}
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*(e*x + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(e*x + d)^(17/2) + 969969*(5*c^3*d^2 - 5*b
*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(15/2) - 373065*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*
e^2 - (b^3 + 6*a*b*c)*e^3)*(e*x + d)^(13/2) + 1322685*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2
- (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(11/2) - 1616615*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e
^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(9/2) + 692835*(
c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a
*b^2 + a^2*c)*d^2*e^4)*(e*x + d)^(7/2))/e^7
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Fricas [B] time = 2.03145, size = 1733, normalized size = 6.06 \begin{align*} \text{result too large to display} \end{align*}
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*(255255*c^3*e^9*x^9 + 5120*c^3*d^9 - 24320*b*c^2*d^8*e - 461890*a^2*b*d^4*e^5 + 692835*a^3*d^3*e^6 +
41344*(b^2*c + a*c^2)*d^7*e^2 - 25840*(b^3 + 6*a*b*c)*d^6*e^3 + 167960*(a*b^2 + a^2*c)*d^5*e^4 + 45045*(13*c^
3*d*e^8 + 19*b*c^2*e^9)*x^8 + 3003*(115*c^3*d^2*e^7 + 665*b*c^2*d*e^8 + 323*(b^2*c + a*c^2)*e^9)*x^7 + 231*(5*
c^3*d^3*e^6 + 5225*b*c^2*d^2*e^7 + 10013*(b^2*c + a*c^2)*d*e^8 + 1615*(b^3 + 6*a*b*c)*e^9)*x^6 - 63*(20*c^3*d^
4*e^5 - 95*b*c^2*d^3*e^6 - 22933*(b^2*c + a*c^2)*d^2*e^7 - 14535*(b^3 + 6*a*b*c)*d*e^8 - 20995*(a*b^2 + a^2*c)
*e^9)*x^5 + 35*(40*c^3*d^5*e^4 - 190*b*c^2*d^4*e^5 + 46189*a^2*b*e^9 + 323*(b^2*c + a*c^2)*d^3*e^6 + 17119*(b^
3 + 6*a*b*c)*d^2*e^7 + 96577*(a*b^2 + a^2*c)*d*e^8)*x^4 - 5*(320*c^3*d^6*e^3 - 1520*b*c^2*d^5*e^4 - 877591*a^2
*b*d*e^8 - 138567*a^3*e^9 + 2584*(b^2*c + a*c^2)*d^4*e^5 - 1615*(b^3 + 6*a*b*c)*d^3*e^6 - 474487*(a*b^2 + a^2*
c)*d^2*e^7)*x^3 + 3*(640*c^3*d^7*e^2 - 3040*b*c^2*d^6*e^3 + 1154725*a^2*b*d^2*e^7 + 692835*a^3*d*e^8 + 5168*(b
^2*c + a*c^2)*d^5*e^4 - 3230*(b^3 + 6*a*b*c)*d^4*e^5 + 20995*(a*b^2 + a^2*c)*d^3*e^6)*x^2 - (2560*c^3*d^8*e -
12160*b*c^2*d^7*e^2 - 230945*a^2*b*d^3*e^6 - 2078505*a^3*d^2*e^7 + 20672*(b^2*c + a*c^2)*d^6*e^3 - 12920*(b^3
+ 6*a*b*c)*d^5*e^4 + 83980*(a*b^2 + a^2*c)*d^4*e^5)*x)*sqrt(e*x + d)/e^7
________________________________________________________________________________________
Sympy [A] time = 65.2801, size = 2363, normalized size = 8.26 \begin{align*} \text{result too large to display} \end{align*}
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] time = 1.23712, size = 2830, normalized size = 9.9 \begin{align*} \text{result too large to display} \end{align*}
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]
2/14549535*(2909907*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b*d^2*e^(-1) + 415701*(15*(x*e + d)^(7/2) -
42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b^2*d^2*e^(-2) + 415701*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5
/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*c*d^2*e^(-2) + 46189*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*
e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b^3*d^2*e^(-3) + 277134*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*
d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*b*c*d^2*e^(-3) + 12597*(315*(x*e + d)^(11/2) - 1540*(
x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^2*c*d^2*e
^(-4) + 12597*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)
*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a*c^2*d^2*e^(-4) + 4845*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 100
10*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b*c^
2*d^2*e^(-5) + 323*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*
e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*c^3*d^2*
e^(-6) + 4849845*(x*e + d)^(3/2)*a^3*d^2 + 831402*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3
/2)*d^2)*a^2*b*d*e^(-1) + 277134*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(
x*e + d)^(3/2)*d^3)*a*b^2*d*e^(-2) + 277134*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*
d^2 - 105*(x*e + d)^(3/2)*d^3)*a^2*c*d*e^(-2) + 8398*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*
e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^3*d*e^(-3) + 50388*(315*(x*e + d)^(1
1/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4
)*a*b*c*d*e^(-3) + 9690*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x
*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b^2*c*d*e^(-4) + 9690*(693*(x*e + d)^
(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2
)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a*c^2*d*e^(-4) + 1938*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61
425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5
+ 15015*(x*e + d)^(3/2)*d^6)*b*c^2*d*e^(-5) + 266*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(
x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 +
153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*c^3*d*e^(-6) + 1939938*(3*(x*e + d)^(5/2) - 5*(x*e + d
)^(3/2)*d)*a^3*d + 138567*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d
)^(3/2)*d^3)*a^2*b*e^(-1) + 12597*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 -
2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a*b^2*e^(-2) + 12597*(315*(x*e + d)^(11/2) - 1540*(x*e +
d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^2*c*e^(-2) + 16
15*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9
009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b^3*e^(-3) + 9690*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(
11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3
/2)*d^5)*a*b*c*e^(-3) + 969*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 1
00100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)
*b^2*c*e^(-4) + 969*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x
*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a*c^2*e
^(-4) + 399*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)
^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*
e + d)^(3/2)*d^7)*b*c^2*e^(-5) + 7*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(1
5/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 83140
20*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*c^3*e^(-6) + 138567*(15*(x*
e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3)*e^(-1)