3.1626 \(\int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)
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))/(2
1*e^7)
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Rubi [A] time = 0.222662, 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
\[ -\frac{12 b^5 (d+e x)^{19/2} (b d-a e)}{19 e^7}+\frac{30 b^4 (d+e x)^{17/2} (b d-a e)^2}{17 e^7}-\frac{8 b^3 (d+e x)^{15/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 (d+e x)^{13/2} (b d-a e)^4}{13 e^7}-\frac{12 b (d+e x)^{11/2} (b d-a e)^5}{11 e^7}+\frac{2 (d+e x)^{9/2} (b d-a e)^6}{9 e^7}+\frac{2 b^6 (d+e x)^{21/2}}{21 e^7} \]
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))/(2
1*e^7)
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Rubi in Sympy [A] time = 84.8264, size = 173, normalized size = 0.93 \[ \frac{2 b^{6} \left (d + e x\right )^{\frac{21}{2}}}{21 e^{7}} + \frac{12 b^{5} \left (d + e x\right )^{\frac{19}{2}} \left (a e - b d\right )}{19 e^{7}} + \frac{30 b^{4} \left (d + e x\right )^{\frac{17}{2}} \left (a e - b d\right )^{2}}{17 e^{7}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )^{3}}{3 e^{7}} + \frac{30 b^{2} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{4}}{13 e^{7}} + \frac{12 b \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{5}}{11 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{6}}{9 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
2*b**6*(d + e*x)**(21/2)/(21*e**7) + 12*b**5*(d + e*x)**(19/2)*(a*e - b*d)/(19*e
**7) + 30*b**4*(d + e*x)**(17/2)*(a*e - b*d)**2/(17*e**7) + 8*b**3*(d + e*x)**(1
5/2)*(a*e - b*d)**3/(3*e**7) + 30*b**2*(d + e*x)**(13/2)*(a*e - b*d)**4/(13*e**7
) + 12*b*(d + e*x)**(11/2)*(a*e - b*d)**5/(11*e**7) + 2*(d + e*x)**(9/2)*(a*e -
b*d)**6/(9*e**7)
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Mathematica [A] time = 0.39017, size = 291, normalized size = 1.56 \[ \frac{2 (d+e x)^{9/2} \left (323323 a^6 e^6+176358 a^5 b e^5 (9 e x-2 d)+33915 a^4 b^2 e^4 \left (8 d^2-36 d e x+99 e^2 x^2\right )+9044 a^3 b^3 e^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+399 a^2 b^4 e^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+42 a b^5 e \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )+b^6 \left (1024 d^6-4608 d^5 e x+12672 d^4 e^2 x^2-27456 d^3 e^3 x^3+51480 d^2 e^4 x^4-87516 d e^5 x^5+138567 e^6 x^6\right )\right )}{2909907 e^7} \]
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*a^6*e^6 + 176358*a^5*b*e^5*(-2*d + 9*e*x) + 33915*a^4
*b^2*e^4*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 9044*a^3*b^3*e^3*(-16*d^3 + 72*d^2*e*
x - 198*d*e^2*x^2 + 429*e^3*x^3) + 399*a^2*b^4*e^2*(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) + 42*a*b^5*e*(-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) + b
^6*(1024*d^6 - 4608*d^5*e*x + 12672*d^4*e^2*x^2 - 27456*d^3*e^3*x^3 + 51480*d^2*
e^4*x^4 - 87516*d*e^5*x^5 + 138567*e^6*x^6)))/(2909907*e^7)
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Maple [B] time = 0.013, size = 377, normalized size = 2. \[{\frac{277134\,{x}^{6}{b}^{6}{e}^{6}+1837836\,{x}^{5}a{b}^{5}{e}^{6}-175032\,{x}^{5}{b}^{6}d{e}^{5}+5135130\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-1081080\,{x}^{4}a{b}^{5}d{e}^{5}+102960\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+7759752\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-2738736\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+576576\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-54912\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+6715170\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-3581424\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+1264032\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-266112\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+25344\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+3174444\,x{a}^{5}b{e}^{6}-2441880\,x{a}^{4}{b}^{2}d{e}^{5}+1302336\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-459648\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+96768\,xa{b}^{5}{d}^{4}{e}^{2}-9216\,x{b}^{6}{d}^{5}e+646646\,{a}^{6}{e}^{6}-705432\,{a}^{5}bd{e}^{5}+542640\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-289408\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+102144\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-21504\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{2909907\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]
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^4-540540*a*b^5*d*e^5*x^4+51480*b^6*d^2*e^4*x^4+387987
6*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^2-133056*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-12209
40*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 [A] time = 0.737313, size = 473, normalized size = 2.53 \[ \frac{2 \,{\left (138567 \,{\left (e x + d\right )}^{\frac{21}{2}} b^{6} - 918918 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{19}{2}} + 2567565 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{17}{2}} - 3879876 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 3357585 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 1587222 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 323323 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{2909907 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(7/2),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) + 33575
85*(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 [A] time = 0.208866, size = 984, normalized size = 5.26 \[ \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}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(7/2),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 = 26.6669, size = 2450, normalized size = 13.1 \[ \text{result too large to display} \]
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)**(1
1/2)/11 + (d + e*x)**(13/2)/13)/e**3 + 40*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)**(15/2)/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) - 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 + 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)/11 - 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/XCAS [A] time = 0.260997, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(7/2),x, algorithm="giac")
[Out]
Done