3.2059 \(\int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)
Optimal. Leaf size=216 \[ -\frac{14 b^6 (d+e x)^{19/2} (b d-a e)}{19 e^8}+\frac{42 b^5 (d+e x)^{17/2} (b d-a e)^2}{17 e^8}-\frac{14 b^4 (d+e x)^{15/2} (b d-a e)^3}{3 e^8}+\frac{70 b^3 (d+e x)^{13/2} (b d-a e)^4}{13 e^8}-\frac{42 b^2 (d+e x)^{11/2} (b d-a e)^5}{11 e^8}+\frac{14 b (d+e x)^{9/2} (b d-a e)^6}{9 e^8}-\frac{2 (d+e x)^{7/2} (b d-a e)^7}{7 e^8}+\frac{2 b^7 (d+e x)^{21/2}}{21 e^8} \]
[Out]
(-2*(b*d - a*e)^7*(d + e*x)^(7/2))/(7*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(9/2)
)/(9*e^8) - (42*b^2*(b*d - a*e)^5*(d + e*x)^(11/2))/(11*e^8) + (70*b^3*(b*d - a*
e)^4*(d + e*x)^(13/2))/(13*e^8) - (14*b^4*(b*d - a*e)^3*(d + e*x)^(15/2))/(3*e^8
) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(17/2))/(17*e^8) - (14*b^6*(b*d - a*e)*(d +
e*x)^(19/2))/(19*e^8) + (2*b^7*(d + e*x)^(21/2))/(21*e^8)
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Rubi [A] time = 0.187887, 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
\[ -\frac{14 b^6 (d+e x)^{19/2} (b d-a e)}{19 e^8}+\frac{42 b^5 (d+e x)^{17/2} (b d-a e)^2}{17 e^8}-\frac{14 b^4 (d+e x)^{15/2} (b d-a e)^3}{3 e^8}+\frac{70 b^3 (d+e x)^{13/2} (b d-a e)^4}{13 e^8}-\frac{42 b^2 (d+e x)^{11/2} (b d-a e)^5}{11 e^8}+\frac{14 b (d+e x)^{9/2} (b d-a e)^6}{9 e^8}-\frac{2 (d+e x)^{7/2} (b d-a e)^7}{7 e^8}+\frac{2 b^7 (d+e x)^{21/2}}{21 e^8} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(-2*(b*d - a*e)^7*(d + e*x)^(7/2))/(7*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(9/2)
)/(9*e^8) - (42*b^2*(b*d - a*e)^5*(d + e*x)^(11/2))/(11*e^8) + (70*b^3*(b*d - a*
e)^4*(d + e*x)^(13/2))/(13*e^8) - (14*b^4*(b*d - a*e)^3*(d + e*x)^(15/2))/(3*e^8
) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(17/2))/(17*e^8) - (14*b^6*(b*d - a*e)*(d +
e*x)^(19/2))/(19*e^8) + (2*b^7*(d + e*x)^(21/2))/(21*e^8)
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Rubi in Sympy [A] time = 109.441, size = 201, normalized size = 0.93 \[ \frac{2 b^{7} \left (d + e x\right )^{\frac{21}{2}}}{21 e^{8}} + \frac{14 b^{6} \left (d + e x\right )^{\frac{19}{2}} \left (a e - b d\right )}{19 e^{8}} + \frac{42 b^{5} \left (d + e x\right )^{\frac{17}{2}} \left (a e - b d\right )^{2}}{17 e^{8}} + \frac{14 b^{4} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )^{3}}{3 e^{8}} + \frac{70 b^{3} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{4}}{13 e^{8}} + \frac{42 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{5}}{11 e^{8}} + \frac{14 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{6}}{9 e^{8}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{7}}{7 e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
2*b**7*(d + e*x)**(21/2)/(21*e**8) + 14*b**6*(d + e*x)**(19/2)*(a*e - b*d)/(19*e
**8) + 42*b**5*(d + e*x)**(17/2)*(a*e - b*d)**2/(17*e**8) + 14*b**4*(d + e*x)**(
15/2)*(a*e - b*d)**3/(3*e**8) + 70*b**3*(d + e*x)**(13/2)*(a*e - b*d)**4/(13*e**
8) + 42*b**2*(d + e*x)**(11/2)*(a*e - b*d)**5/(11*e**8) + 14*b*(d + e*x)**(9/2)*
(a*e - b*d)**6/(9*e**8) + 2*(d + e*x)**(7/2)*(a*e - b*d)**7/(7*e**8)
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Mathematica [A] time = 0.613494, size = 376, normalized size = 1.74 \[ \frac{2 (d+e x)^{7/2} \left (415701 a^7 e^7+323323 a^6 b e^6 (7 e x-2 d)+88179 a^5 b^2 e^5 \left (8 d^2-28 d e x+63 e^2 x^2\right )+33915 a^4 b^3 e^4 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+2261 a^3 b^4 e^3 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+399 a^2 b^5 e^2 \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+21 a b^6 e \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )+b^7 \left (-2048 d^7+7168 d^6 e x-16128 d^5 e^2 x^2+29568 d^4 e^3 x^3-48048 d^3 e^4 x^4+72072 d^2 e^5 x^5-102102 d e^6 x^6+138567 e^7 x^7\right )\right )}{2909907 e^8} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(7/2)*(415701*a^7*e^7 + 323323*a^6*b*e^6*(-2*d + 7*e*x) + 88179*a^5
*b^2*e^5*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 33915*a^4*b^3*e^4*(-16*d^3 + 56*d^2*e
*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 2261*a^3*b^4*e^3*(128*d^4 - 448*d^3*e*x + 10
08*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 399*a^2*b^5*e^2*(-256*d^5 + 89
6*d^4*e*x - 2016*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5)
+ 21*a*b^6*e*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x^2 - 14784*d^3*e^3*x^3 +
24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6) + b^7*(-2048*d^7 + 7168*d^6
*e*x - 16128*d^5*e^2*x^2 + 29568*d^4*e^3*x^3 - 48048*d^3*e^4*x^4 + 72072*d^2*e^5
*x^5 - 102102*d*e^6*x^6 + 138567*e^7*x^7)))/(2909907*e^8)
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Maple [B] time = 0.013, size = 498, normalized size = 2.3 \[{\frac{277134\,{b}^{7}{x}^{7}{e}^{7}+2144142\,a{b}^{6}{e}^{7}{x}^{6}-204204\,{b}^{7}d{e}^{6}{x}^{6}+7189182\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}-1513512\,a{b}^{6}d{e}^{6}{x}^{5}+144144\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}+13579566\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}-4792788\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}+1009008\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}-96096\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}+15668730\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}-8356656\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}+2949408\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}-620928\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}+59136\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}+11110554\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}-8546580\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}+4558176\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}-1608768\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}+338688\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}-32256\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}+4526522\,{a}^{6}b{e}^{7}x-4938024\,{a}^{5}{b}^{2}d{e}^{6}x+3798480\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x-2025856\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x+715008\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x-150528\,a{b}^{6}{d}^{5}{e}^{2}x+14336\,{b}^{7}{d}^{6}ex+831402\,{a}^{7}{e}^{7}-1293292\,{a}^{6}bd{e}^{6}+1410864\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}-1085280\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}+578816\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}-204288\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}+43008\,a{b}^{6}{d}^{6}e-4096\,{b}^{7}{d}^{7}}{2909907\,{e}^{8}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/2909907*(e*x+d)^(7/2)*(138567*b^7*e^7*x^7+1072071*a*b^6*e^7*x^6-102102*b^7*d*e
^6*x^6+3594591*a^2*b^5*e^7*x^5-756756*a*b^6*d*e^6*x^5+72072*b^7*d^2*e^5*x^5+6789
783*a^3*b^4*e^7*x^4-2396394*a^2*b^5*d*e^6*x^4+504504*a*b^6*d^2*e^5*x^4-48048*b^7
*d^3*e^4*x^4+7834365*a^4*b^3*e^7*x^3-4178328*a^3*b^4*d*e^6*x^3+1474704*a^2*b^5*d
^2*e^5*x^3-310464*a*b^6*d^3*e^4*x^3+29568*b^7*d^4*e^3*x^3+5555277*a^5*b^2*e^7*x^
2-4273290*a^4*b^3*d*e^6*x^2+2279088*a^3*b^4*d^2*e^5*x^2-804384*a^2*b^5*d^3*e^4*x
^2+169344*a*b^6*d^4*e^3*x^2-16128*b^7*d^5*e^2*x^2+2263261*a^6*b*e^7*x-2469012*a^
5*b^2*d*e^6*x+1899240*a^4*b^3*d^2*e^5*x-1012928*a^3*b^4*d^3*e^4*x+357504*a^2*b^5
*d^4*e^3*x-75264*a*b^6*d^5*e^2*x+7168*b^7*d^6*e*x+415701*a^7*e^7-646646*a^6*b*d*
e^6+705432*a^5*b^2*d^2*e^5-542640*a^4*b^3*d^3*e^4+289408*a^3*b^4*d^4*e^3-102144*
a^2*b^5*d^5*e^2+21504*a*b^6*d^6*e-2048*b^7*d^7)/e^8
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Maxima [A] time = 0.730486, size = 616, normalized size = 2.85 \[ \frac{2 \,{\left (138567 \,{\left (e x + d\right )}^{\frac{21}{2}} b^{7} - 1072071 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{19}{2}} + 3594591 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{17}{2}} - 6789783 \,{\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 7834365 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 5555277 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 2263261 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 415701 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{2909907 \, e^{8}} \]
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)^(5/2),x, algorithm="maxima")
[Out]
2/2909907*(138567*(e*x + d)^(21/2)*b^7 - 1072071*(b^7*d - a*b^6*e)*(e*x + d)^(19
/2) + 3594591*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(17/2) - 6789783*(
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)^(15/2) + 7834
365*(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)^(13/2) - 5555277*(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)^(11/2) + 2263261*(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)^(9/2) - 415701*(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)^(7/2))/e^8
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Fricas [A] time = 0.280772, size = 1057, normalized size = 4.89 \[ \text{result too large to display} \]
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)^(5/2),x, algorithm="fricas")
[Out]
2/2909907*(138567*b^7*e^10*x^10 - 2048*b^7*d^10 + 21504*a*b^6*d^9*e - 102144*a^2
*b^5*d^8*e^2 + 289408*a^3*b^4*d^7*e^3 - 542640*a^4*b^3*d^6*e^4 + 705432*a^5*b^2*
d^5*e^5 - 646646*a^6*b*d^4*e^6 + 415701*a^7*d^3*e^7 + 7293*(43*b^7*d*e^9 + 147*a
*b^6*e^10)*x^9 + 3861*(47*b^7*d^2*e^8 + 637*a*b^6*d*e^9 + 931*a^2*b^5*e^10)*x^8
+ 429*(b^7*d^3*e^7 + 3381*a*b^6*d^2*e^8 + 19551*a^2*b^5*d*e^9 + 15827*a^3*b^4*e^
10)*x^7 - 231*(2*b^7*d^4*e^6 - 21*a*b^6*d^3*e^7 - 21945*a^2*b^5*d^2*e^8 - 70091*
a^3*b^4*d*e^9 - 33915*a^4*b^3*e^10)*x^6 + 63*(8*b^7*d^5*e^5 - 84*a*b^6*d^4*e^6 +
399*a^2*b^5*d^3*e^7 + 160531*a^3*b^4*d^2*e^8 + 305235*a^4*b^3*d*e^9 + 88179*a^5
*b^2*e^10)*x^5 - 7*(80*b^7*d^6*e^4 - 840*a*b^6*d^5*e^5 + 3990*a^2*b^5*d^4*e^6 -
11305*a^3*b^4*d^3*e^7 - 1797495*a^4*b^3*d^2*e^8 - 2028117*a^5*b^2*d*e^9 - 323323
*a^6*b*e^10)*x^4 + (640*b^7*d^7*e^3 - 6720*a*b^6*d^6*e^4 + 31920*a^2*b^5*d^5*e^5
- 90440*a^3*b^4*d^4*e^6 + 169575*a^4*b^3*d^3*e^7 + 9964227*a^5*b^2*d^2*e^8 + 61
43137*a^6*b*d*e^9 + 415701*a^7*e^10)*x^3 - 3*(256*b^7*d^8*e^2 - 2688*a*b^6*d^7*e
^3 + 12768*a^2*b^5*d^6*e^4 - 36176*a^3*b^4*d^5*e^5 + 67830*a^4*b^3*d^4*e^6 - 881
79*a^5*b^2*d^3*e^7 - 1616615*a^6*b*d^2*e^8 - 415701*a^7*d*e^9)*x^2 + (1024*b^7*d
^9*e - 10752*a*b^6*d^8*e^2 + 51072*a^2*b^5*d^7*e^3 - 144704*a^3*b^4*d^6*e^4 + 27
1320*a^4*b^3*d^5*e^5 - 352716*a^5*b^2*d^4*e^6 + 323323*a^6*b*d^3*e^7 + 1247103*a
^7*d^2*e^8)*x)*sqrt(e*x + d)/e^8
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Sympy [A] time = 23.5756, size = 2096, normalized size = 9.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
a**7*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4
*a**7*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*a**7*(d**2*(d + e*x)*
*(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e + 14*a**6*b*d**2*(-d*(
d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 28*a**6*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 + 14*a**6*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 + 42*a**5*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 + 84*a**5*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 + 42*a
**5*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 + 70*a**4*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 + 140*a**4*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 + 70*a**4*b**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**4 + 70*a**3*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 + 140*a**3*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 + 70*a**3*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 + 42*a**2*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 + 84*a**2*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)/1
3 + (d + e*x)**(15/2)/15)/e**6 + 42*a**2*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 + 14*a*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)/1
5)/e**7 + 28*a*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)/1
1 + 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 + 14*a*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 + 2*b**7*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**8 + 4*b**7*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**8 + 2*b**7*(-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
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.356189, 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)^(5/2),x, algorithm="giac")
[Out]
Done