3.1638 \(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=185 \[ -\frac{12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac{30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac{4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac{2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac{2 b^6 (d+e x)^{19/2}}{19 e^7} \]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(7/2))/(7*e^7) - (4*b*(b*d - a*e)^5*(d + e*x)^(9/2))/(3*e^7) + (30*b^2*(b*d - a*e)^
4*(d + e*x)^(11/2))/(11*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(13/2))/(13*e^7) + (2*b^4*(b*d - a*e)^2*(d + e*
x)^(15/2))/e^7 - (12*b^5*(b*d - a*e)*(d + e*x)^(17/2))/(17*e^7) + (2*b^6*(d + e*x)^(19/2))/(19*e^7)

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Rubi [A]  time = 0.0608597, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {27, 43} \[ -\frac{12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac{30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac{4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac{2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac{2 b^6 (d+e x)^{19/2}}{19 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(7/2))/(7*e^7) - (4*b*(b*d - a*e)^5*(d + e*x)^(9/2))/(3*e^7) + (30*b^2*(b*d - a*e)^
4*(d + e*x)^(11/2))/(11*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(13/2))/(13*e^7) + (2*b^4*(b*d - a*e)^2*(d + e*
x)^(15/2))/e^7 - (12*b^5*(b*d - a*e)*(d + e*x)^(17/2))/(17*e^7) + (2*b^6*(d + e*x)^(19/2))/(19*e^7)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^6 (d+e x)^{5/2}}{e^6}-\frac{6 b (b d-a e)^5 (d+e x)^{7/2}}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{9/2}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{11/2}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{13/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{15/2}}{e^6}+\frac{b^6 (d+e x)^{17/2}}{e^6}\right ) \, dx\\ &=\frac{2 (b d-a e)^6 (d+e x)^{7/2}}{7 e^7}-\frac{4 b (b d-a e)^5 (d+e x)^{9/2}}{3 e^7}+\frac{30 b^2 (b d-a e)^4 (d+e x)^{11/2}}{11 e^7}-\frac{40 b^3 (b d-a e)^3 (d+e x)^{13/2}}{13 e^7}+\frac{2 b^4 (b d-a e)^2 (d+e x)^{15/2}}{e^7}-\frac{12 b^5 (b d-a e) (d+e x)^{17/2}}{17 e^7}+\frac{2 b^6 (d+e x)^{19/2}}{19 e^7}\\ \end{align*}

Mathematica [A]  time = 0.112956, size = 145, normalized size = 0.78 \[ \frac{2 (d+e x)^{7/2} \left (1322685 b^2 (d+e x)^2 (b d-a e)^4-1492260 b^3 (d+e x)^3 (b d-a e)^3+969969 b^4 (d+e x)^4 (b d-a e)^2-342342 b^5 (d+e x)^5 (b d-a e)-646646 b (d+e x) (b d-a e)^5+138567 (b d-a e)^6+51051 b^6 (d+e x)^6\right )}{969969 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(138567*(b*d - a*e)^6 - 646646*b*(b*d - a*e)^5*(d + e*x) + 1322685*b^2*(b*d - a*e)^4*(d + e
*x)^2 - 1492260*b^3*(b*d - a*e)^3*(d + e*x)^3 + 969969*b^4*(b*d - a*e)^2*(d + e*x)^4 - 342342*b^5*(b*d - a*e)*
(d + e*x)^5 + 51051*b^6*(d + e*x)^6))/(969969*e^7)

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Maple [B]  time = 0.046, size = 377, normalized size = 2. \begin{align*}{\frac{102102\,{b}^{6}{x}^{6}{e}^{6}+684684\,{x}^{5}a{b}^{5}{e}^{6}-72072\,{x}^{5}{b}^{6}d{e}^{5}+1939938\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-456456\,{x}^{4}a{b}^{5}d{e}^{5}+48048\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+2984520\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-1193808\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+280896\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-29568\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+2645370\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-1627920\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+651168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-153216\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+16128\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+1293292\,x{a}^{5}b{e}^{6}-1175720\,x{a}^{4}{b}^{2}d{e}^{5}+723520\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-289408\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+68096\,xa{b}^{5}{d}^{4}{e}^{2}-7168\,x{b}^{6}{d}^{5}e+277134\,{a}^{6}{e}^{6}-369512\,{a}^{5}bd{e}^{5}+335920\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-206720\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}+82688\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-19456\,a{b}^{5}{d}^{5}e+2048\,{d}^{6}{b}^{6}}{969969\,{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)*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2/969969*(e*x+d)^(7/2)*(51051*b^6*e^6*x^6+342342*a*b^5*e^6*x^5-36036*b^6*d*e^5*x^5+969969*a^2*b^4*e^6*x^4-2282
28*a*b^5*d*e^5*x^4+24024*b^6*d^2*e^4*x^4+1492260*a^3*b^3*e^6*x^3-596904*a^2*b^4*d*e^5*x^3+140448*a*b^5*d^2*e^4
*x^3-14784*b^6*d^3*e^3*x^3+1322685*a^4*b^2*e^6*x^2-813960*a^3*b^3*d*e^5*x^2+325584*a^2*b^4*d^2*e^4*x^2-76608*a
*b^5*d^3*e^3*x^2+8064*b^6*d^4*e^2*x^2+646646*a^5*b*e^6*x-587860*a^4*b^2*d*e^5*x+361760*a^3*b^3*d^2*e^4*x-14470
4*a^2*b^4*d^3*e^3*x+34048*a*b^5*d^4*e^2*x-3584*b^6*d^5*e*x+138567*a^6*e^6-184756*a^5*b*d*e^5+167960*a^4*b^2*d^
2*e^4-103360*a^3*b^3*d^3*e^3+41344*a^2*b^4*d^4*e^2-9728*a*b^5*d^5*e+1024*b^6*d^6)/e^7

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Maxima [B]  time = 1.06292, size = 473, normalized size = 2.56 \begin{align*} \frac{2 \,{\left (51051 \,{\left (e x + d\right )}^{\frac{19}{2}} b^{6} - 342342 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 969969 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 1492260 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 646646 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 138567 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{969969 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/969969*(51051*(e*x + d)^(19/2)*b^6 - 342342*(b^6*d - a*b^5*e)*(e*x + d)^(17/2) + 969969*(b^6*d^2 - 2*a*b^5*d
*e + a^2*b^4*e^2)*(e*x + d)^(15/2) - 1492260*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x +
d)^(13/2) + 1322685*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*(e*x + d)^(1
1/2) - 646646*(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5
)*(e*x + d)^(9/2) + 138567*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2
*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*(e*x + d)^(7/2))/e^7

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Fricas [B]  time = 1.65116, size = 1497, normalized size = 8.09 \begin{align*} \frac{2 \,{\left (51051 \, b^{6} e^{9} x^{9} + 1024 \, b^{6} d^{9} - 9728 \, a b^{5} d^{8} e + 41344 \, a^{2} b^{4} d^{7} e^{2} - 103360 \, a^{3} b^{3} d^{6} e^{3} + 167960 \, a^{4} b^{2} d^{5} e^{4} - 184756 \, a^{5} b d^{4} e^{5} + 138567 \, a^{6} d^{3} e^{6} + 9009 \,{\left (13 \, b^{6} d e^{8} + 38 \, a b^{5} e^{9}\right )} x^{8} + 3003 \,{\left (23 \, b^{6} d^{2} e^{7} + 266 \, a b^{5} d e^{8} + 323 \, a^{2} b^{4} e^{9}\right )} x^{7} + 231 \,{\left (b^{6} d^{3} e^{6} + 2090 \, a b^{5} d^{2} e^{7} + 10013 \, a^{2} b^{4} d e^{8} + 6460 \, a^{3} b^{3} e^{9}\right )} x^{6} - 63 \,{\left (4 \, b^{6} d^{4} e^{5} - 38 \, a b^{5} d^{3} e^{6} - 22933 \, a^{2} b^{4} d^{2} e^{7} - 58140 \, a^{3} b^{3} d e^{8} - 20995 \, a^{4} b^{2} e^{9}\right )} x^{5} + 7 \,{\left (40 \, b^{6} d^{5} e^{4} - 380 \, a b^{5} d^{4} e^{5} + 1615 \, a^{2} b^{4} d^{3} e^{6} + 342380 \, a^{3} b^{3} d^{2} e^{7} + 482885 \, a^{4} b^{2} d e^{8} + 92378 \, a^{5} b e^{9}\right )} x^{4} -{\left (320 \, b^{6} d^{6} e^{3} - 3040 \, a b^{5} d^{5} e^{4} + 12920 \, a^{2} b^{4} d^{4} e^{5} - 32300 \, a^{3} b^{3} d^{3} e^{6} - 2372435 \, a^{4} b^{2} d^{2} e^{7} - 1755182 \, a^{5} b d e^{8} - 138567 \, a^{6} e^{9}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{7} e^{2} - 1216 \, a b^{5} d^{6} e^{3} + 5168 \, a^{2} b^{4} d^{5} e^{4} - 12920 \, a^{3} b^{3} d^{4} e^{5} + 20995 \, a^{4} b^{2} d^{3} e^{6} + 461890 \, a^{5} b d^{2} e^{7} + 138567 \, a^{6} d e^{8}\right )} x^{2} -{\left (512 \, b^{6} d^{8} e - 4864 \, a b^{5} d^{7} e^{2} + 20672 \, a^{2} b^{4} d^{6} e^{3} - 51680 \, a^{3} b^{3} d^{5} e^{4} + 83980 \, a^{4} b^{2} d^{4} e^{5} - 92378 \, a^{5} b d^{3} e^{6} - 415701 \, a^{6} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{969969 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/969969*(51051*b^6*e^9*x^9 + 1024*b^6*d^9 - 9728*a*b^5*d^8*e + 41344*a^2*b^4*d^7*e^2 - 103360*a^3*b^3*d^6*e^3
+ 167960*a^4*b^2*d^5*e^4 - 184756*a^5*b*d^4*e^5 + 138567*a^6*d^3*e^6 + 9009*(13*b^6*d*e^8 + 38*a*b^5*e^9)*x^8
+ 3003*(23*b^6*d^2*e^7 + 266*a*b^5*d*e^8 + 323*a^2*b^4*e^9)*x^7 + 231*(b^6*d^3*e^6 + 2090*a*b^5*d^2*e^7 + 100
13*a^2*b^4*d*e^8 + 6460*a^3*b^3*e^9)*x^6 - 63*(4*b^6*d^4*e^5 - 38*a*b^5*d^3*e^6 - 22933*a^2*b^4*d^2*e^7 - 5814
0*a^3*b^3*d*e^8 - 20995*a^4*b^2*e^9)*x^5 + 7*(40*b^6*d^5*e^4 - 380*a*b^5*d^4*e^5 + 1615*a^2*b^4*d^3*e^6 + 3423
80*a^3*b^3*d^2*e^7 + 482885*a^4*b^2*d*e^8 + 92378*a^5*b*e^9)*x^4 - (320*b^6*d^6*e^3 - 3040*a*b^5*d^5*e^4 + 129
20*a^2*b^4*d^4*e^5 - 32300*a^3*b^3*d^3*e^6 - 2372435*a^4*b^2*d^2*e^7 - 1755182*a^5*b*d*e^8 - 138567*a^6*e^9)*x
^3 + 3*(128*b^6*d^7*e^2 - 1216*a*b^5*d^6*e^3 + 5168*a^2*b^4*d^5*e^4 - 12920*a^3*b^3*d^4*e^5 + 20995*a^4*b^2*d^
3*e^6 + 461890*a^5*b*d^2*e^7 + 138567*a^6*d*e^8)*x^2 - (512*b^6*d^8*e - 4864*a*b^5*d^7*e^2 + 20672*a^2*b^4*d^6
*e^3 - 51680*a^3*b^3*d^5*e^4 + 83980*a^4*b^2*d^4*e^5 - 92378*a^5*b*d^3*e^6 - 415701*a^6*d^2*e^7)*x)*sqrt(e*x +
d)/e^7

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Sympy [A]  time = 51.2127, size = 1671, normalized size = 9.03 \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)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**6*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**6*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
12*a**5*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 24*a**5*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 + 12*a**5*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 + 30*a**4*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 + 60*a**4*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 + 30*a**4*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 + 40*a**3*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
+ 80*a**3*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 + 40*a**3*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 1
0*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
+ 30*a**2*b**4*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 + 60*a**2*b**4*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 + 30*a**2*b**4*(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
+ 12*a*b**5*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 + 24*a*b**5*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 + 12*a*b**5*(-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*b**6*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*b**6*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*b**6*(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 = 1.3244, size = 2007, normalized size = 10.85 \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)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

2/14549535*(5819814*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^5*b*d^2*e^(-1) + 2078505*(15*(x*e + d)^(7/2) -
42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^4*b^2*d^2*e^(-2) + 923780*(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^3*b^3*d^2*e^(-3) + 62985*(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*b^4*d^2*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*b^5*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)*b^6*d^2*e^(-6) + 4849845*(x*e + d)^(3/2)
*a^6*d^2 + 1662804*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*b*d*e^(-1) + 13856
70*(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^4*b^2*d*
e^(-2) + 167960*(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^3*b^3*d*e^(-3) + 48450*(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
^2*b^4*d*e^(-4) + 3876*(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*b^
5*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 - 3646
5*(x*e + d)^(3/2)*d^7)*b^6*d*e^(-6) + 1939938*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6*d + 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^5*b*e^(-1) + 62985
*(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^4*b^2*e^(-2) + 32300*(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^3*b^3*e^(-3) + 4
845*(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^2*b^4*e^(-4) + 798*(6
435*(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)*a*b^5*e^(-5) + 7*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 895
3560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(
7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*b^6*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^6)*e^(-1)