3.1637 \(\int (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=187 \[ -\frac{12 b^5 (d+e x)^{19/2} (b d-a e)}{19 e^7}+\frac{30 b^4 (d+e x)^{17/2} (b d-a e)^2}{17 e^7}-\frac{8 b^3 (d+e x)^{15/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 (d+e x)^{13/2} (b d-a e)^4}{13 e^7}-\frac{12 b (d+e x)^{11/2} (b d-a e)^5}{11 e^7}+\frac{2 (d+e x)^{9/2} (b d-a e)^6}{9 e^7}+\frac{2 b^6 (d+e x)^{21/2}}{21 e^7} \]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(9/2))/(9*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(11/2))/(11*e^7) + (30*b^2*(b*d - a*
e)^4*(d + e*x)^(13/2))/(13*e^7) - (8*b^3*(b*d - a*e)^3*(d + e*x)^(15/2))/(3*e^7) + (30*b^4*(b*d - a*e)^2*(d +
e*x)^(17/2))/(17*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(19/2))/(19*e^7) + (2*b^6*(d + e*x)^(21/2))/(21*e^7)

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Rubi [A]  time = 0.082821, antiderivative size = 187, 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} \[ -\frac{12 b^5 (d+e x)^{19/2} (b d-a e)}{19 e^7}+\frac{30 b^4 (d+e x)^{17/2} (b d-a e)^2}{17 e^7}-\frac{8 b^3 (d+e x)^{15/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 (d+e x)^{13/2} (b d-a e)^4}{13 e^7}-\frac{12 b (d+e x)^{11/2} (b d-a e)^5}{11 e^7}+\frac{2 (d+e x)^{9/2} (b d-a e)^6}{9 e^7}+\frac{2 b^6 (d+e x)^{21/2}}{21 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(9/2))/(9*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(11/2))/(11*e^7) + (30*b^2*(b*d - a*
e)^4*(d + e*x)^(13/2))/(13*e^7) - (8*b^3*(b*d - a*e)^3*(d + e*x)^(15/2))/(3*e^7) + (30*b^4*(b*d - a*e)^2*(d +
e*x)^(17/2))/(17*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(19/2))/(19*e^7) + (2*b^6*(d + e*x)^(21/2))/(21*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)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^6 (d+e x)^{7/2}}{e^6}-\frac{6 b (b d-a e)^5 (d+e x)^{9/2}}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{11/2}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{13/2}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{15/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{17/2}}{e^6}+\frac{b^6 (d+e x)^{19/2}}{e^6}\right ) \, dx\\ &=\frac{2 (b d-a e)^6 (d+e x)^{9/2}}{9 e^7}-\frac{12 b (b d-a e)^5 (d+e x)^{11/2}}{11 e^7}+\frac{30 b^2 (b d-a e)^4 (d+e x)^{13/2}}{13 e^7}-\frac{8 b^3 (b d-a e)^3 (d+e x)^{15/2}}{3 e^7}+\frac{30 b^4 (b d-a e)^2 (d+e x)^{17/2}}{17 e^7}-\frac{12 b^5 (b d-a e) (d+e x)^{19/2}}{19 e^7}+\frac{2 b^6 (d+e x)^{21/2}}{21 e^7}\\ \end{align*}

Mathematica [A]  time = 0.158601, size = 145, normalized size = 0.78 \[ \frac{2 (d+e x)^{9/2} \left (3357585 b^2 (d+e x)^2 (b d-a e)^4-3879876 b^3 (d+e x)^3 (b d-a e)^3+2567565 b^4 (d+e x)^4 (b d-a e)^2-918918 b^5 (d+e x)^5 (b d-a e)-1587222 b (d+e x) (b d-a e)^5+323323 (b d-a e)^6+138567 b^6 (d+e x)^6\right )}{2909907 e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(d + e*x)^(9/2)*(323323*(b*d - a*e)^6 - 1587222*b*(b*d - a*e)^5*(d + e*x) + 3357585*b^2*(b*d - a*e)^4*(d +
e*x)^2 - 3879876*b^3*(b*d - a*e)^3*(d + e*x)^3 + 2567565*b^4*(b*d - a*e)^2*(d + e*x)^4 - 918918*b^5*(b*d - a*e
)*(d + e*x)^5 + 138567*b^6*(d + e*x)^6))/(2909907*e^7)

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Maple [B]  time = 0.049, size = 377, normalized size = 2. \begin{align*}{\frac{277134\,{b}^{6}{x}^{6}{e}^{6}+1837836\,{x}^{5}a{b}^{5}{e}^{6}-175032\,{x}^{5}{b}^{6}d{e}^{5}+5135130\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-1081080\,{x}^{4}a{b}^{5}d{e}^{5}+102960\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+7759752\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-2738736\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+576576\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-54912\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+6715170\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-3581424\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+1264032\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-266112\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+25344\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+3174444\,x{a}^{5}b{e}^{6}-2441880\,x{a}^{4}{b}^{2}d{e}^{5}+1302336\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-459648\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+96768\,xa{b}^{5}{d}^{4}{e}^{2}-9216\,x{b}^{6}{d}^{5}e+646646\,{a}^{6}{e}^{6}-705432\,{a}^{5}bd{e}^{5}+542640\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-289408\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}+102144\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-21504\,a{b}^{5}{d}^{5}e+2048\,{d}^{6}{b}^{6}}{2909907\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2/2909907*(e*x+d)^(9/2)*(138567*b^6*e^6*x^6+918918*a*b^5*e^6*x^5-87516*b^6*d*e^5*x^5+2567565*a^2*b^4*e^6*x^4-5
40540*a*b^5*d*e^5*x^4+51480*b^6*d^2*e^4*x^4+3879876*a^3*b^3*e^6*x^3-1369368*a^2*b^4*d*e^5*x^3+288288*a*b^5*d^2
*e^4*x^3-27456*b^6*d^3*e^3*x^3+3357585*a^4*b^2*e^6*x^2-1790712*a^3*b^3*d*e^5*x^2+632016*a^2*b^4*d^2*e^4*x^2-13
3056*a*b^5*d^3*e^3*x^2+12672*b^6*d^4*e^2*x^2+1587222*a^5*b*e^6*x-1220940*a^4*b^2*d*e^5*x+651168*a^3*b^3*d^2*e^
4*x-229824*a^2*b^4*d^3*e^3*x+48384*a*b^5*d^4*e^2*x-4608*b^6*d^5*e*x+323323*a^6*e^6-352716*a^5*b*d*e^5+271320*a
^4*b^2*d^2*e^4-144704*a^3*b^3*d^3*e^3+51072*a^2*b^4*d^4*e^2-10752*a*b^5*d^5*e+1024*b^6*d^6)/e^7

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Maxima [B]  time = 1.06954, size = 473, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (138567 \,{\left (e x + d\right )}^{\frac{21}{2}} b^{6} - 918918 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{19}{2}} + 2567565 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{17}{2}} - 3879876 \,{\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{15}{2}} + 3357585 \,{\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{13}{2}} - 1587222 \,{\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{11}{2}} + 323323 \,{\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{9}{2}}\right )}}{2909907 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

2/2909907*(138567*(e*x + d)^(21/2)*b^6 - 918918*(b^6*d - a*b^5*e)*(e*x + d)^(19/2) + 2567565*(b^6*d^2 - 2*a*b^
5*d*e + a^2*b^4*e^2)*(e*x + d)^(17/2) - 3879876*(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)^(15/2) + 3357585*(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)
^(13/2) - 1587222*(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)^(11/2) + 323323*(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)^(9/2))/e^7

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Fricas [B]  time = 1.60444, size = 1746, normalized size = 9.34 \begin{align*} \frac{2 \,{\left (138567 \, b^{6} e^{10} x^{10} + 1024 \, b^{6} d^{10} - 10752 \, a b^{5} d^{9} e + 51072 \, a^{2} b^{4} d^{8} e^{2} - 144704 \, a^{3} b^{3} d^{7} e^{3} + 271320 \, a^{4} b^{2} d^{6} e^{4} - 352716 \, a^{5} b d^{5} e^{5} + 323323 \, a^{6} d^{4} e^{6} + 14586 \,{\left (32 \, b^{6} d e^{9} + 63 \, a b^{5} e^{10}\right )} x^{9} + 3861 \,{\left (138 \, b^{6} d^{2} e^{8} + 812 \, a b^{5} d e^{9} + 665 \, a^{2} b^{4} e^{10}\right )} x^{8} + 1716 \,{\left (121 \, b^{6} d^{3} e^{7} + 2121 \, a b^{5} d^{2} e^{8} + 5187 \, a^{2} b^{4} d e^{9} + 2261 \, a^{3} b^{3} e^{10}\right )} x^{7} + 231 \,{\left (b^{6} d^{4} e^{6} + 6288 \, a b^{5} d^{3} e^{7} + 45714 \, a^{2} b^{4} d^{2} e^{8} + 59432 \, a^{3} b^{3} d e^{9} + 14535 \, a^{4} b^{2} e^{10}\right )} x^{6} - 126 \,{\left (2 \, b^{6} d^{5} e^{5} - 21 \, a b^{5} d^{4} e^{6} - 34542 \, a^{2} b^{4} d^{3} e^{7} - 133076 \, a^{3} b^{3} d^{2} e^{8} - 96900 \, a^{4} b^{2} d e^{9} - 12597 \, a^{5} b e^{10}\right )} x^{5} + 7 \,{\left (40 \, b^{6} d^{6} e^{4} - 420 \, a b^{5} d^{5} e^{5} + 1995 \, a^{2} b^{4} d^{4} e^{6} + 1033600 \, a^{3} b^{3} d^{3} e^{7} + 2219010 \, a^{4} b^{2} d^{2} e^{8} + 856596 \, a^{5} b d e^{9} + 46189 \, a^{6} e^{10}\right )} x^{4} - 4 \,{\left (80 \, b^{6} d^{7} e^{3} - 840 \, a b^{5} d^{6} e^{4} + 3990 \, a^{2} b^{4} d^{5} e^{5} - 11305 \, a^{3} b^{3} d^{4} e^{6} - 1797495 \, a^{4} b^{2} d^{3} e^{7} - 2028117 \, a^{5} b d^{2} e^{8} - 323323 \, a^{6} d e^{9}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{8} e^{2} - 1344 \, a b^{5} d^{7} e^{3} + 6384 \, a^{2} b^{4} d^{6} e^{4} - 18088 \, a^{3} b^{3} d^{5} e^{5} + 33915 \, a^{4} b^{2} d^{4} e^{6} + 1410864 \, a^{5} b d^{3} e^{7} + 646646 \, a^{6} d^{2} e^{8}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{9} e - 2688 \, a b^{5} d^{8} e^{2} + 12768 \, a^{2} b^{4} d^{7} e^{3} - 36176 \, a^{3} b^{3} d^{6} e^{4} + 67830 \, a^{4} b^{2} d^{5} e^{5} - 88179 \, a^{5} b d^{4} e^{6} - 646646 \, a^{6} d^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{2909907 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

2/2909907*(138567*b^6*e^10*x^10 + 1024*b^6*d^10 - 10752*a*b^5*d^9*e + 51072*a^2*b^4*d^8*e^2 - 144704*a^3*b^3*d
^7*e^3 + 271320*a^4*b^2*d^6*e^4 - 352716*a^5*b*d^5*e^5 + 323323*a^6*d^4*e^6 + 14586*(32*b^6*d*e^9 + 63*a*b^5*e
^10)*x^9 + 3861*(138*b^6*d^2*e^8 + 812*a*b^5*d*e^9 + 665*a^2*b^4*e^10)*x^8 + 1716*(121*b^6*d^3*e^7 + 2121*a*b^
5*d^2*e^8 + 5187*a^2*b^4*d*e^9 + 2261*a^3*b^3*e^10)*x^7 + 231*(b^6*d^4*e^6 + 6288*a*b^5*d^3*e^7 + 45714*a^2*b^
4*d^2*e^8 + 59432*a^3*b^3*d*e^9 + 14535*a^4*b^2*e^10)*x^6 - 126*(2*b^6*d^5*e^5 - 21*a*b^5*d^4*e^6 - 34542*a^2*
b^4*d^3*e^7 - 133076*a^3*b^3*d^2*e^8 - 96900*a^4*b^2*d*e^9 - 12597*a^5*b*e^10)*x^5 + 7*(40*b^6*d^6*e^4 - 420*a
*b^5*d^5*e^5 + 1995*a^2*b^4*d^4*e^6 + 1033600*a^3*b^3*d^3*e^7 + 2219010*a^4*b^2*d^2*e^8 + 856596*a^5*b*d*e^9 +
46189*a^6*e^10)*x^4 - 4*(80*b^6*d^7*e^3 - 840*a*b^5*d^6*e^4 + 3990*a^2*b^4*d^5*e^5 - 11305*a^3*b^3*d^4*e^6 -
1797495*a^4*b^2*d^3*e^7 - 2028117*a^5*b*d^2*e^8 - 323323*a^6*d*e^9)*x^3 + 3*(128*b^6*d^8*e^2 - 1344*a*b^5*d^7*
e^3 + 6384*a^2*b^4*d^6*e^4 - 18088*a^3*b^3*d^5*e^5 + 33915*a^4*b^2*d^4*e^6 + 1410864*a^5*b*d^3*e^7 + 646646*a^
6*d^2*e^8)*x^2 - 2*(256*b^6*d^9*e - 2688*a*b^5*d^8*e^2 + 12768*a^2*b^4*d^7*e^3 - 36176*a^3*b^3*d^6*e^4 + 67830
*a^4*b^2*d^5*e^5 - 88179*a^5*b*d^4*e^6 - 646646*a^6*d^3*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 78.9065, size = 2450, normalized size = 13.1 \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)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d**3*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 6*a**6*d**2*(-d*(d + e*x)**(3/2
)/3 + (d + e*x)**(5/2)/5)/e + 6*a**6*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*a**6*(-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 + 12*a**5*b*d**3*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 36*a**5*b*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**2 + 36*a**5*b*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**2 + 12*a**5*b*(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**2 + 30*a*
*4*b**2*d**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 90*a**4*b**2*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**3 + 90
*a**4*b**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**3 + 30*a**4*b**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**3 + 4
0*a**3*b**3*d**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**4 + 120*a**3*b**3*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**4 + 120*a**3*b**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**4 + 40*a**3*b**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)**(1
5/2)/15)/e**4 + 30*a**2*b**4*d**3*(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 + 90*a**2*b**4*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**5 + 90*a**2*b**4*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**5 + 30*a**2*b**4*(-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**5 + 12*a*b**5*d**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**6
+ 36*a*b**5*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**6 + 36
*a*b**5*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**6 + 12*a*b**5*(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**6 + 2*b**6*d**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)/1
1 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 6*b**6*d**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**7 + 6*b**6*d*(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 + 2*b**6*(-d**9*(d + e*x)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5 - 36*d**7*(d + e*x)**(7/2)/7 +
28*d**6*(d + e*x)**(9/2)/3 - 126*d**5*(d + e*x)**(11/2)/11 + 126*d**4*(d + e*x)**(13/2)/13 - 28*d**3*(d + e*x
)**(15/2)/5 + 36*d**2*(d + e*x)**(17/2)/17 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x)**(21/2)/21)/e**7

________________________________________________________________________________________

Giac [B]  time = 1.36297, size = 2936, normalized size = 15.7 \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)^(7/2)*(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^3*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^3*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^3*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^3*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^3*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^3*e^(-6) + 4849845*(x*e + d)^(3/2)
*a^6*d^3 + 2494206*(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^2*e^(-1) + 207
8505*(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^2*e^(-2) + 251940*(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^2*e^(-3) + 72675*(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^2*e^(-4) + 5814*(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^2*e^(-5) + 399*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 3
48075*(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^6*d^2*e^(-6) + 2909907*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6*d^2 +
831402*(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*
d*e^(-1) + 188955*(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*d*e^(-2) + 96900*(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*d*e^(-3) + 14535*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100
100*(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*d*e^(-4) + 2394*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 34807
5*(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*d*e^(-5) + 21*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 387987
6*(x*e + d)^(15/2)*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 + 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*d*e^(-6)
+ 415701*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^6*d + 25194*(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^5*b*e^(-1) + 24225*(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^4*b^2*e^(-2) + 6460*(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^3*b^3*e^(-3) + 1995*(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)*a^2*b^4*e^(-4) +
42*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(1
3/2)*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 - 332560
8*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*a*b^5*e^(-5) + 3*(230945*(x*e + d)^(21/2) - 2297295*(x*e +
d)^(19/2)*d + 10270260*(x*e + d)^(17/2)*d^2 - 27159132*(x*e + d)^(15/2)*d^3 + 47006190*(x*e + d)^(13/2)*d^4 -
55552770*(x*e + d)^(11/2)*d^5 + 45265220*(x*e + d)^(9/2)*d^6 - 24942060*(x*e + d)^(7/2)*d^7 + 8729721*(x*e +
d)^(5/2)*d^8 - 1616615*(x*e + d)^(3/2)*d^9)*b^6*e^(-6) + 46189*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 1
89*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^6)*e^(-1)