3.2058 \(\int (a+b x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
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)
________________________________________________________________________________________
Rubi [A] time = 0.108485, 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, Rules used =
{27, 43} \[ -\frac{2 b^6 (d+e x)^{21/2} (b d-a e)}{3 e^8}+\frac{42 b^5 (d+e x)^{19/2} (b d-a e)^2}{19 e^8}-\frac{70 b^4 (d+e x)^{17/2} (b d-a e)^3}{17 e^8}+\frac{14 b^3 (d+e x)^{15/2} (b d-a e)^4}{3 e^8}-\frac{42 b^2 (d+e x)^{13/2} (b d-a e)^5}{13 e^8}+\frac{14 b (d+e x)^{11/2} (b d-a e)^6}{11 e^8}-\frac{2 (d+e x)^{9/2} (b d-a e)^7}{9 e^8}+\frac{2 b^7 (d+e x)^{23/2}}{23 e^8} \]
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)
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 (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^7 (d+e x)^{7/2}}{e^7}+\frac{7 b (b d-a e)^6 (d+e x)^{9/2}}{e^7}-\frac{21 b^2 (b d-a e)^5 (d+e x)^{11/2}}{e^7}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{13/2}}{e^7}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{15/2}}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{17/2}}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^{19/2}}{e^7}+\frac{b^7 (d+e x)^{21/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (b d-a e)^7 (d+e x)^{9/2}}{9 e^8}+\frac{14 b (b d-a e)^6 (d+e x)^{11/2}}{11 e^8}-\frac{42 b^2 (b d-a e)^5 (d+e x)^{13/2}}{13 e^8}+\frac{14 b^3 (b d-a e)^4 (d+e x)^{15/2}}{3 e^8}-\frac{70 b^4 (b d-a e)^3 (d+e x)^{17/2}}{17 e^8}+\frac{42 b^5 (b d-a e)^2 (d+e x)^{19/2}}{19 e^8}-\frac{2 b^6 (b d-a e) (d+e x)^{21/2}}{3 e^8}+\frac{2 b^7 (d+e x)^{23/2}}{23 e^8}\\ \end{align*}
Mathematica [A] time = 0.202042, size = 167, normalized size = 0.77 \[ \frac{2 (d+e x)^{9/2} \left (-15444891 b^2 (d+e x)^2 (b d-a e)^5+22309287 b^3 (d+e x)^3 (b d-a e)^4-19684665 b^4 (d+e x)^4 (b d-a e)^3+10567557 b^5 (d+e x)^5 (b d-a e)^2-3187041 b^6 (d+e x)^6 (b d-a e)+6084351 b (d+e x) (b d-a e)^6-1062347 (b d-a e)^7+415701 b^7 (d+e x)^7\right )}{9561123 e^8} \]
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*(b*d - a*e)^7 + 6084351*b*(b*d - a*e)^6*(d + e*x) - 15444891*b^2*(b*d - a*e)^5*(d
+ e*x)^2 + 22309287*b^3*(b*d - a*e)^4*(d + e*x)^3 - 19684665*b^4*(b*d - a*e)^3*(d + e*x)^4 + 10567557*b^5*(b*
d - a*e)^2*(d + e*x)^5 - 3187041*b^6*(b*d - a*e)*(d + e*x)^6 + 415701*b^7*(d + e*x)^7))/(9561123*e^8)
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Maple [B] time = 0.006, size = 498, normalized size = 2.3 \begin{align*}{\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}}}} \end{align*}
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+19684665*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^4-102960*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+48
45456*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+556416*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^7-1352078*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 [B] time = 1.04367, size = 616, normalized size = 2.85 \begin{align*} \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}} \end{align*}
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, 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) + 22309287*(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 + 1
5*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
________________________________________________________________________________________
Fricas [B] time = 1.37876, size = 2188, normalized size = 10.13 \begin{align*} \text{result too large to display} \end{align*}
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, 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 + 13
8567*(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 + 153
0374*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 - 71452122*a^5*b^2*d^2*e^9 - 229853
26*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 - 1223
60*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 + 4249388*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 = 82.5452, size = 3046, normalized size = 14.1 \begin{align*} \text{result too large to display} \end{align*}
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 + 1
26*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)/1
3)/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 + 126*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 + 2
8*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)**(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**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 - 25
2*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 [B] time = 1.32289, size = 3645, normalized size = 16.88 \begin{align*} \text{result too large to display} \end{align*}
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, algorithm="giac")
[Out]
2/334639305*(156165009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6*b*d^3*e^(-1) + 66927861*(15*(x*e + d)^(7/
2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*b^2*d^3*e^(-2) + 37182145*(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^3*d^3*e^(-3) + 3380195*(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^4*d^3*e^(-4) + 780045*(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^5*d^3*e^(-5) + 52003
*(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^6*d^3*e^(-6) + 3059*(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)*b^7*d^3*e^(-7) + 111546435*(x*e + d)^(3/2)*a^7*d^3 + 66927861*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d +
35*(x*e + d)^(3/2)*d^2)*a^6*b*d^2*e^(-1) + 66927861*(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^2*d^2*e^(-2) + 10140585*(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^3*d^2*e^(-3) +
3900225*(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^4*d^2*e^(-4) + 468027*(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^5*d^2*e^(-5) + 64239*(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*b^6*d^2*e^(-6) +
483*(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)*b^7*d^2*e^(-7) + 66927861*(3*(x*e + d)^(5/2) - 5*(x*e + d)
^(3/2)*d)*a^7*d^2 + 22309287*(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^6*b*d*e^(-1) + 6084351*(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^2*d*e^(-2) + 3900225*(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^3*d*e^(-3) + 780045*(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 + 15
015*(x*e + d)^(3/2)*d^6)*a^3*b^4*d*e^(-4) + 192717*(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^5*d*e^(-5) + 3381*(109395*(x*e + d)^(19/2) - 97
8120*(x*e + d)^(17/2)*d + 3879876*(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)*a*b^6*d*e^(-6) + 207*(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^7*d*e^(-7) + 9561123*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^7
*d + 676039*(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^6*b*e^(-1) + 780045*(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^5*b^2*
e^(-2) + 260015*(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^4*b^3*e^(
-3) + 107065*(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^3*b^4*e^(-4) + 3381*(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)^(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)*a^2*b^5*e^(-5) + 483*
(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 - 24942
060*(x*e + d)^(7/2)*d^7 + 8729721*(x*e + d)^(5/2)*d^8 - 1616615*(x*e + d)^(3/2)*d^9)*a*b^6*e^(-6) + 15*(969969
*(x*e + d)^(23/2) - 10623470*(x*e + d)^(21/2)*d + 52837785*(x*e + d)^(19/2)*d^2 - 157477320*(x*e + d)^(17/2)*d
^3 + 312330018*(x*e + d)^(15/2)*d^4 - 432456948*(x*e + d)^(13/2)*d^5 + 425904570*(x*e + d)^(11/2)*d^6 - 297457
160*(x*e + d)^(9/2)*d^7 + 143416845*(x*e + d)^(7/2)*d^8 - 44618574*(x*e + d)^(5/2)*d^9 + 7436429*(x*e + d)^(3/
2)*d^10)*b^7*e^(-7) + 1062347*(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^7)*e^(-1)