3.2058 \(\int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)
Optimal. Leaf size=216 \[ -\frac{2 b^6 (d+e x)^{21/2} (b d-a e)}{3 e^8}+\frac{42 b^5 (d+e x)^{19/2} (b d-a e)^2}{19 e^8}-\frac{70 b^4 (d+e x)^{17/2} (b d-a e)^3}{17 e^8}+\frac{14 b^3 (d+e x)^{15/2} (b d-a e)^4}{3 e^8}-\frac{42 b^2 (d+e x)^{13/2} (b d-a e)^5}{13 e^8}+\frac{14 b (d+e x)^{11/2} (b d-a e)^6}{11 e^8}-\frac{2 (d+e x)^{9/2} (b d-a e)^7}{9 e^8}+\frac{2 b^7 (d+e x)^{23/2}}{23 e^8} \]
[Out]
(-2*(b*d - a*e)^7*(d + e*x)^(9/2))/(9*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(11/2
))/(11*e^8) - (42*b^2*(b*d - a*e)^5*(d + e*x)^(13/2))/(13*e^8) + (14*b^3*(b*d -
a*e)^4*(d + e*x)^(15/2))/(3*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(17/2))/(17*e
^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(19/2))/(19*e^8) - (2*b^6*(b*d - a*e)*(d +
e*x)^(21/2))/(3*e^8) + (2*b^7*(d + e*x)^(23/2))/(23*e^8)
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Rubi [A] time = 0.242027, antiderivative size = 216, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061
\[ -\frac{2 b^6 (d+e x)^{21/2} (b d-a e)}{3 e^8}+\frac{42 b^5 (d+e x)^{19/2} (b d-a e)^2}{19 e^8}-\frac{70 b^4 (d+e x)^{17/2} (b d-a e)^3}{17 e^8}+\frac{14 b^3 (d+e x)^{15/2} (b d-a e)^4}{3 e^8}-\frac{42 b^2 (d+e x)^{13/2} (b d-a e)^5}{13 e^8}+\frac{14 b (d+e x)^{11/2} (b d-a e)^6}{11 e^8}-\frac{2 (d+e x)^{9/2} (b d-a e)^7}{9 e^8}+\frac{2 b^7 (d+e x)^{23/2}}{23 e^8} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(-2*(b*d - a*e)^7*(d + e*x)^(9/2))/(9*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(11/2
))/(11*e^8) - (42*b^2*(b*d - a*e)^5*(d + e*x)^(13/2))/(13*e^8) + (14*b^3*(b*d -
a*e)^4*(d + e*x)^(15/2))/(3*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(17/2))/(17*e
^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(19/2))/(19*e^8) - (2*b^6*(b*d - a*e)*(d +
e*x)^(21/2))/(3*e^8) + (2*b^7*(d + e*x)^(23/2))/(23*e^8)
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Rubi in Sympy [A] time = 110.127, size = 201, normalized size = 0.93 \[ \frac{2 b^{7} \left (d + e x\right )^{\frac{23}{2}}}{23 e^{8}} + \frac{2 b^{6} \left (d + e x\right )^{\frac{21}{2}} \left (a e - b d\right )}{3 e^{8}} + \frac{42 b^{5} \left (d + e x\right )^{\frac{19}{2}} \left (a e - b d\right )^{2}}{19 e^{8}} + \frac{70 b^{4} \left (d + e x\right )^{\frac{17}{2}} \left (a e - b d\right )^{3}}{17 e^{8}} + \frac{14 b^{3} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )^{4}}{3 e^{8}} + \frac{42 b^{2} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{5}}{13 e^{8}} + \frac{14 b \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{6}}{11 e^{8}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{7}}{9 e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
2*b**7*(d + e*x)**(23/2)/(23*e**8) + 2*b**6*(d + e*x)**(21/2)*(a*e - b*d)/(3*e**
8) + 42*b**5*(d + e*x)**(19/2)*(a*e - b*d)**2/(19*e**8) + 70*b**4*(d + e*x)**(17
/2)*(a*e - b*d)**3/(17*e**8) + 14*b**3*(d + e*x)**(15/2)*(a*e - b*d)**4/(3*e**8)
+ 42*b**2*(d + e*x)**(13/2)*(a*e - b*d)**5/(13*e**8) + 14*b*(d + e*x)**(11/2)*(
a*e - b*d)**6/(11*e**8) + 2*(d + e*x)**(9/2)*(a*e - b*d)**7/(9*e**8)
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Mathematica [A] time = 0.603081, size = 376, normalized size = 1.74 \[ \frac{2 (d+e x)^{9/2} \left (1062347 a^7 e^7+676039 a^6 b e^6 (9 e x-2 d)+156009 a^5 b^2 e^5 \left (8 d^2-36 d e x+99 e^2 x^2\right )+52003 a^4 b^3 e^4 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+3059 a^3 b^4 e^3 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+483 a^2 b^5 e^2 \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )+23 a b^6 e \left (1024 d^6-4608 d^5 e x+12672 d^4 e^2 x^2-27456 d^3 e^3 x^3+51480 d^2 e^4 x^4-87516 d e^5 x^5+138567 e^6 x^6\right )+b^7 \left (-2048 d^7+9216 d^6 e x-25344 d^5 e^2 x^2+54912 d^4 e^3 x^3-102960 d^3 e^4 x^4+175032 d^2 e^5 x^5-277134 d e^6 x^6+415701 e^7 x^7\right )\right )}{9561123 e^8} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(9/2)*(1062347*a^7*e^7 + 676039*a^6*b*e^6*(-2*d + 9*e*x) + 156009*a
^5*b^2*e^5*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 52003*a^4*b^3*e^4*(-16*d^3 + 72*d^2
*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 3059*a^3*b^4*e^3*(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) + 483*a^2*b^5*e^2*(-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) + 23*a*b^6*e*(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) + b^7*(-2048*d^7 + 9
216*d^6*e*x - 25344*d^5*e^2*x^2 + 54912*d^4*e^3*x^3 - 102960*d^3*e^4*x^4 + 17503
2*d^2*e^5*x^5 - 277134*d*e^6*x^6 + 415701*e^7*x^7)))/(9561123*e^8)
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Maple [B] time = 0.012, size = 498, normalized size = 2.3 \[{\frac{831402\,{b}^{7}{x}^{7}{e}^{7}+6374082\,a{b}^{6}{e}^{7}{x}^{6}-554268\,{b}^{7}d{e}^{6}{x}^{6}+21135114\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}-4025736\,a{b}^{6}d{e}^{6}{x}^{5}+350064\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}+39369330\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}-12432420\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}+2368080\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}-205920\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}+44618574\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}-20996976\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}+6630624\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}-1262976\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}+109824\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}+30889782\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}-20593188\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}+9690912\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}-3060288\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}+582912\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}-50688\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}+12168702\,{a}^{6}b{e}^{7}x-11232648\,{a}^{5}{b}^{2}d{e}^{6}x+7488432\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x-3523968\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x+1112832\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x-211968\,a{b}^{6}{d}^{5}{e}^{2}x+18432\,{b}^{7}{d}^{6}ex+2124694\,{a}^{7}{e}^{7}-2704156\,{a}^{6}bd{e}^{6}+2496144\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}-1664096\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}+783104\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}-247296\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}+47104\,a{b}^{6}{d}^{6}e-4096\,{b}^{7}{d}^{7}}{9561123\,{e}^{8}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/9561123*(e*x+d)^(9/2)*(415701*b^7*e^7*x^7+3187041*a*b^6*e^7*x^6-277134*b^7*d*e
^6*x^6+10567557*a^2*b^5*e^7*x^5-2012868*a*b^6*d*e^6*x^5+175032*b^7*d^2*e^5*x^5+1
9684665*a^3*b^4*e^7*x^4-6216210*a^2*b^5*d*e^6*x^4+1184040*a*b^6*d^2*e^5*x^4-1029
60*b^7*d^3*e^4*x^4+22309287*a^4*b^3*e^7*x^3-10498488*a^3*b^4*d*e^6*x^3+3315312*a
^2*b^5*d^2*e^5*x^3-631488*a*b^6*d^3*e^4*x^3+54912*b^7*d^4*e^3*x^3+15444891*a^5*b
^2*e^7*x^2-10296594*a^4*b^3*d*e^6*x^2+4845456*a^3*b^4*d^2*e^5*x^2-1530144*a^2*b^
5*d^3*e^4*x^2+291456*a*b^6*d^4*e^3*x^2-25344*b^7*d^5*e^2*x^2+6084351*a^6*b*e^7*x
-5616324*a^5*b^2*d*e^6*x+3744216*a^4*b^3*d^2*e^5*x-1761984*a^3*b^4*d^3*e^4*x+556
416*a^2*b^5*d^4*e^3*x-105984*a*b^6*d^5*e^2*x+9216*b^7*d^6*e*x+1062347*a^7*e^7-13
52078*a^6*b*d*e^6+1248072*a^5*b^2*d^2*e^5-832048*a^4*b^3*d^3*e^4+391552*a^3*b^4*
d^4*e^3-123648*a^2*b^5*d^5*e^2+23552*a*b^6*d^6*e-2048*b^7*d^7)/e^8
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Maxima [A] time = 0.753434, size = 616, normalized size = 2.85 \[ \frac{2 \,{\left (415701 \,{\left (e x + d\right )}^{\frac{23}{2}} b^{7} - 3187041 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{21}{2}} + 10567557 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{19}{2}} - 19684665 \,{\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 22309287 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 15444891 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 6084351 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 1062347 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{9561123 \, e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
2/9561123*(415701*(e*x + d)^(23/2)*b^7 - 3187041*(b^7*d - a*b^6*e)*(e*x + d)^(21
/2) + 10567557*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(19/2) - 19684665
*(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(e*x + d)^(17/2) + 22
309287*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*
e^4)*(e*x + d)^(15/2) - 15444891*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 -
10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*b^2*e^5)*(e*x + d)^(13/2) + 6084351*
(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*
d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e*x + d)^(11/2) - 1062347*(b^7*d^7 - 7*a
*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a
^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(e*x + d)^(9/2))/e^8
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Fricas [A] time = 0.284915, size = 1203, normalized size = 5.57 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
2/9561123*(415701*b^7*e^11*x^11 - 2048*b^7*d^11 + 23552*a*b^6*d^10*e - 123648*a^
2*b^5*d^9*e^2 + 391552*a^3*b^4*d^8*e^3 - 832048*a^4*b^3*d^7*e^4 + 1248072*a^5*b^
2*d^6*e^5 - 1352078*a^6*b*d^5*e^6 + 1062347*a^7*d^4*e^7 + 138567*(10*b^7*d*e^10
+ 23*a*b^6*e^11)*x^10 + 7293*(214*b^7*d^2*e^9 + 1472*a*b^6*d*e^10 + 1449*a^2*b^5
*e^11)*x^9 + 1287*(464*b^7*d^3*e^8 + 9522*a*b^6*d^2*e^9 + 28014*a^2*b^5*d*e^10 +
15295*a^3*b^4*e^11)*x^8 + 429*(b^7*d^4*e^7 + 11132*a*b^6*d^3*e^8 + 97566*a^2*b^
5*d^2*e^9 + 159068*a^3*b^4*d*e^10 + 52003*a^4*b^3*e^11)*x^7 - 231*(2*b^7*d^5*e^6
- 23*a*b^6*d^4*e^7 - 72312*a^2*b^5*d^3*e^8 - 350474*a^3*b^4*d^2*e^9 - 341734*a^
4*b^3*d*e^10 - 66861*a^5*b^2*e^11)*x^6 + 63*(8*b^7*d^6*e^5 - 92*a*b^6*d^5*e^6 +
483*a^2*b^5*d^4*e^7 + 529644*a^3*b^4*d^3*e^8 + 1530374*a^4*b^3*d^2*e^9 + 891480*
a^5*b^2*d*e^10 + 96577*a^6*b*e^11)*x^5 - (560*b^7*d^7*e^4 - 6440*a*b^6*d^6*e^5 +
33810*a^2*b^5*d^5*e^6 - 107065*a^3*b^4*d^4*e^7 - 41602400*a^4*b^3*d^3*e^8 - 714
52122*a^5*b^2*d^2*e^9 - 22985326*a^6*b*d*e^10 - 1062347*a^7*e^11)*x^4 + (640*b^7
*d^8*e^3 - 7360*a*b^6*d^7*e^4 + 38640*a^2*b^5*d^6*e^5 - 122360*a^3*b^4*d^5*e^6 +
260015*a^4*b^3*d^4*e^7 + 33073908*a^5*b^2*d^3*e^8 + 31097794*a^6*b*d^2*e^9 + 42
49388*a^7*d*e^10)*x^3 - 3*(256*b^7*d^9*e^2 - 2944*a*b^6*d^8*e^3 + 15456*a^2*b^5*
d^7*e^4 - 48944*a^3*b^4*d^6*e^5 + 104006*a^4*b^3*d^5*e^6 - 156009*a^5*b^2*d^4*e^
7 - 5408312*a^6*b*d^3*e^8 - 2124694*a^7*d^2*e^9)*x^2 + (1024*b^7*d^10*e - 11776*
a*b^6*d^9*e^2 + 61824*a^2*b^5*d^8*e^3 - 195776*a^3*b^4*d^7*e^4 + 416024*a^4*b^3*
d^6*e^5 - 624036*a^5*b^2*d^5*e^6 + 676039*a^6*b*d^4*e^7 + 4249388*a^7*d^3*e^8)*x
)*sqrt(e*x + d)/e^8
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Sympy [A] time = 33.6285, size = 3046, normalized size = 14.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
a**7*d**3*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 6
*a**7*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 6*a**7*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**7*(-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 + 14*a**6*b*d**3*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**
2 + 42*a**6*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 + 42*a**6*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 + 14*a**6*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 + 42*a**5*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 + 126*a**5*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 + 126*a**5*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 + 42*a**5*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 + 70*a**4*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 + 210*a**4*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
+ 210*a**4*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 + 70*a**4*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 + 70
*a**3*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 + 210*a**
3*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 + 210*a**3*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 + 70*a**
3*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 + 42*a
**2*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 + 126*a**2*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 + 1
26*a**2*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 + 42*a**2*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 + 14*a*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 + 42*a*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 + 42*a*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 + 14*a*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 + 2*b**7*d**3*(-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**8 + 6*b**7*d**2*(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**8 + 6*b**7*d*(-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)**(1
3/2)/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**8 + 2*b**7*(d**10*(d + e*x)**(3/2)/3
- 2*d**9*(d + e*x)**(5/2) + 45*d**8*(d + e*x)**(7/2)/7 - 40*d**7*(d + e*x)**(9/2
)/3 + 210*d**6*(d + e*x)**(11/2)/11 - 252*d**5*(d + e*x)**(13/2)/13 + 14*d**4*(d
+ e*x)**(15/2) - 120*d**3*(d + e*x)**(17/2)/17 + 45*d**2*(d + e*x)**(19/2)/19 -
10*d*(d + e*x)**(21/2)/21 + (d + e*x)**(23/2)/23)/e**8
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.374806, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*(e*x + d)^(7/2),x, algorithm="giac")
[Out]
Done