3.1793 \(\int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)
Optimal. Leaf size=218 \[ -\frac{2 b^3 (d+e x)^{15/2} (-4 a B e-A b e+5 b B d)}{15 e^6}+\frac{4 b^2 (d+e x)^{13/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{13 e^6}-\frac{4 b (d+e x)^{11/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{11 e^6}+\frac{2 (d+e x)^{9/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6}-\frac{2 (d+e x)^{7/2} (b d-a e)^4 (B d-A e)}{7 e^6}+\frac{2 b^4 B (d+e x)^{17/2}}{17 e^6} \]
[Out]
(-2*(b*d - a*e)^4*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^6) + (2*(b*d - a*e)^3*(5*b*B
*d - 4*A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3
*A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^6) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A
*b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*
(d + e*x)^(15/2))/(15*e^6) + (2*b^4*B*(d + e*x)^(17/2))/(17*e^6)
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Rubi [A] time = 0.275126, antiderivative size = 218, 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
\[ -\frac{2 b^3 (d+e x)^{15/2} (-4 a B e-A b e+5 b B d)}{15 e^6}+\frac{4 b^2 (d+e x)^{13/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{13 e^6}-\frac{4 b (d+e x)^{11/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{11 e^6}+\frac{2 (d+e x)^{9/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6}-\frac{2 (d+e x)^{7/2} (b d-a e)^4 (B d-A e)}{7 e^6}+\frac{2 b^4 B (d+e x)^{17/2}}{17 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(-2*(b*d - a*e)^4*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^6) + (2*(b*d - a*e)^3*(5*b*B
*d - 4*A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3
*A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^6) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A
*b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*
(d + e*x)^(15/2))/(15*e^6) + (2*b^4*B*(d + e*x)^(17/2))/(17*e^6)
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Rubi in Sympy [A] time = 108.921, size = 221, normalized size = 1.01 \[ \frac{2 B b^{4} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{6}} + \frac{2 b^{3} \left (d + e x\right )^{\frac{15}{2}} \left (A b e + 4 B a e - 5 B b d\right )}{15 e^{6}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{13 e^{6}} + \frac{4 b \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{11 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{9 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{7 e^{6}} \]
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)**2,x)
[Out]
2*B*b**4*(d + e*x)**(17/2)/(17*e**6) + 2*b**3*(d + e*x)**(15/2)*(A*b*e + 4*B*a*e
- 5*B*b*d)/(15*e**6) + 4*b**2*(d + e*x)**(13/2)*(a*e - b*d)*(2*A*b*e + 3*B*a*e
- 5*B*b*d)/(13*e**6) + 4*b*(d + e*x)**(11/2)*(a*e - b*d)**2*(3*A*b*e + 2*B*a*e -
5*B*b*d)/(11*e**6) + 2*(d + e*x)**(9/2)*(a*e - b*d)**3*(4*A*b*e + B*a*e - 5*B*b
*d)/(9*e**6) + 2*(d + e*x)**(7/2)*(A*e - B*d)*(a*e - b*d)**4/(7*e**6)
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Mathematica [A] time = 0.739371, size = 340, normalized size = 1.56 \[ \frac{2 (d+e x)^{7/2} \left (12155 a^4 e^4 (9 A e-2 B d+7 B e x)+4420 a^3 b e^3 \left (11 A e (7 e x-2 d)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-510 a^2 b^2 e^2 \left (3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+68 a b^3 e \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+b^4 \left (17 A e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )-5 B \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{765765 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(2*(d + e*x)^(7/2)*(12155*a^4*e^4*(-2*B*d + 9*A*e + 7*B*e*x) + 4420*a^3*b*e^3*(1
1*A*e*(-2*d + 7*e*x) + B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 510*a^2*b^2*e^2*(-13
*A*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3*B*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2
- 231*e^3*x^3)) + 68*a*b^3*e*(15*A*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231
*e^3*x^3) + B*(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)) + b^4*(17*A*e*(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) - 5*B*(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))))/(765765*e^6)
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Maple [B] time = 0.016, size = 469, normalized size = 2.2 \[{\frac{90090\,{b}^{4}B{x}^{5}{e}^{5}+102102\,A{b}^{4}{e}^{5}{x}^{4}+408408\,Ba{b}^{3}{e}^{5}{x}^{4}-60060\,B{b}^{4}d{e}^{4}{x}^{4}+471240\,Aa{b}^{3}{e}^{5}{x}^{3}-62832\,A{b}^{4}d{e}^{4}{x}^{3}+706860\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-251328\,Ba{b}^{3}d{e}^{4}{x}^{3}+36960\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+835380\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-257040\,Aa{b}^{3}d{e}^{4}{x}^{2}+34272\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+556920\,B{a}^{3}b{e}^{5}{x}^{2}-385560\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+137088\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-20160\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+680680\,A{a}^{3}b{e}^{5}x-371280\,A{a}^{2}{b}^{2}d{e}^{4}x+114240\,Aa{b}^{3}{d}^{2}{e}^{3}x-15232\,A{b}^{4}{d}^{3}{e}^{2}x+170170\,B{a}^{4}{e}^{5}x-247520\,B{a}^{3}bd{e}^{4}x+171360\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-60928\,Ba{b}^{3}{d}^{3}{e}^{2}x+8960\,B{b}^{4}{d}^{4}ex+218790\,A{a}^{4}{e}^{5}-194480\,Ad{a}^{3}b{e}^{4}+106080\,A{a}^{2}{b}^{2}{d}^{2}{e}^{3}-32640\,Aa{b}^{3}{d}^{3}{e}^{2}+4352\,A{d}^{4}{b}^{4}e-48620\,B{a}^{4}d{e}^{4}+70720\,B{d}^{2}{a}^{3}b{e}^{3}-48960\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+17408\,B{d}^{4}a{b}^{3}e-2560\,{b}^{4}B{d}^{5}}{765765\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
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)^2,x)
[Out]
2/765765*(e*x+d)^(7/2)*(45045*B*b^4*e^5*x^5+51051*A*b^4*e^5*x^4+204204*B*a*b^3*e
^5*x^4-30030*B*b^4*d*e^4*x^4+235620*A*a*b^3*e^5*x^3-31416*A*b^4*d*e^4*x^3+353430
*B*a^2*b^2*e^5*x^3-125664*B*a*b^3*d*e^4*x^3+18480*B*b^4*d^2*e^3*x^3+417690*A*a^2
*b^2*e^5*x^2-128520*A*a*b^3*d*e^4*x^2+17136*A*b^4*d^2*e^3*x^2+278460*B*a^3*b*e^5
*x^2-192780*B*a^2*b^2*d*e^4*x^2+68544*B*a*b^3*d^2*e^3*x^2-10080*B*b^4*d^3*e^2*x^
2+340340*A*a^3*b*e^5*x-185640*A*a^2*b^2*d*e^4*x+57120*A*a*b^3*d^2*e^3*x-7616*A*b
^4*d^3*e^2*x+85085*B*a^4*e^5*x-123760*B*a^3*b*d*e^4*x+85680*B*a^2*b^2*d^2*e^3*x-
30464*B*a*b^3*d^3*e^2*x+4480*B*b^4*d^4*e*x+109395*A*a^4*e^5-97240*A*a^3*b*d*e^4+
53040*A*a^2*b^2*d^2*e^3-16320*A*a*b^3*d^3*e^2+2176*A*b^4*d^4*e-24310*B*a^4*d*e^4
+35360*B*a^3*b*d^2*e^3-24480*B*a^2*b^2*d^3*e^2+8704*B*a*b^3*d^4*e-1280*B*b^4*d^5
)/e^6
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Maxima [A] time = 0.743099, size = 552, normalized size = 2.53 \[ \frac{2 \,{\left (45045 \,{\left (e x + d\right )}^{\frac{17}{2}} B b^{4} - 51051 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 117810 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 139230 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 109395 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{765765 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
2/765765*(45045*(e*x + d)^(17/2)*B*b^4 - 51051*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*
e)*(e*x + d)^(15/2) + 117810*(5*B*b^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d*e + (3*B*a^2
*b^2 + 2*A*a*b^3)*e^2)*(e*x + d)^(13/2) - 139230*(5*B*b^4*d^3 - 3*(4*B*a*b^3 + A
*b^4)*d^2*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a^3*b + 3*A*a^2*b^2)*e^3)
*(e*x + d)^(11/2) + 85085*(5*B*b^4*d^4 - 4*(4*B*a*b^3 + A*b^4)*d^3*e + 6*(3*B*a^
2*b^2 + 2*A*a*b^3)*d^2*e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^
3*b)*e^4)*(e*x + d)^(9/2) - 109395*(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*
d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^
3 + (B*a^4 + 4*A*a^3*b)*d*e^4)*(e*x + d)^(7/2))/e^6
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Fricas [A] time = 0.290938, size = 1042, normalized size = 4.78 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
2/765765*(45045*B*b^4*e^8*x^8 - 1280*B*b^4*d^8 + 109395*A*a^4*d^3*e^5 + 2176*(4*
B*a*b^3 + A*b^4)*d^7*e - 8160*(3*B*a^2*b^2 + 2*A*a*b^3)*d^6*e^2 + 17680*(2*B*a^3
*b + 3*A*a^2*b^2)*d^5*e^3 - 24310*(B*a^4 + 4*A*a^3*b)*d^4*e^4 + 3003*(35*B*b^4*d
*e^7 + 17*(4*B*a*b^3 + A*b^4)*e^8)*x^7 + 231*(275*B*b^4*d^2*e^6 + 527*(4*B*a*b^3
+ A*b^4)*d*e^7 + 510*(3*B*a^2*b^2 + 2*A*a*b^3)*e^8)*x^6 + 63*(5*B*b^4*d^3*e^5 +
1207*(4*B*a*b^3 + A*b^4)*d^2*e^6 + 4590*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^7 + 2210*
(2*B*a^3*b + 3*A*a^2*b^2)*e^8)*x^5 - 35*(10*B*b^4*d^4*e^4 - 17*(4*B*a*b^3 + A*b^
4)*d^3*e^5 - 5406*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^6 - 10166*(2*B*a^3*b + 3*A*a^2
*b^2)*d*e^7 - 2431*(B*a^4 + 4*A*a^3*b)*e^8)*x^4 + 5*(80*B*b^4*d^5*e^3 + 21879*A*
a^4*e^8 - 136*(4*B*a*b^3 + A*b^4)*d^4*e^4 + 510*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^
5 + 49946*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^6 + 46189*(B*a^4 + 4*A*a^3*b)*d*e^7)*x
^3 - 3*(160*B*b^4*d^6*e^2 - 109395*A*a^4*d*e^7 - 272*(4*B*a*b^3 + A*b^4)*d^5*e^3
+ 1020*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^4 - 2210*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e
^5 - 60775*(B*a^4 + 4*A*a^3*b)*d^2*e^6)*x^2 + (640*B*b^4*d^7*e + 328185*A*a^4*d^
2*e^6 - 1088*(4*B*a*b^3 + A*b^4)*d^6*e^2 + 4080*(3*B*a^2*b^2 + 2*A*a*b^3)*d^5*e^
3 - 8840*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^4 + 12155*(B*a^4 + 4*A*a^3*b)*d^3*e^5)*
x)*sqrt(e*x + d)/e^6
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Sympy [A] time = 24.2119, size = 2193, normalized size = 10.06 \[ \text{result too large to display} \]
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)**2,x)
[Out]
A*a**4*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) +
4*A*a**4*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*A*a**4*(d**2*(d +
e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e + 8*A*a**3*b*d**
2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 16*A*a**3*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 + 8*A*a**3*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 + 12*A*a**2*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 + 24*A*a**2*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 + 12*A*a**2*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
+ 8*A*a*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 + 16*A*a*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 + 8*A*a*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 + 2*A*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 + 4*A*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 + 2*A*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 + 2*B*a**4*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)*
*(5/2)/5)/e**2 + 4*B*a**4*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 + 2*B*a**4*(-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 + 8*B*a**3*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**3 + 1
6*B*a**3*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**3 + 8*B*a**3*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**3 + 12*B*a**2*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**4 + 24*
B*a**2*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**4 + 12*B*a**2
*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**4 + 8*B*a*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*
*5 + 16*B*a*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**5 + 8*B*a*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**5 + 2*B*
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)**(1
3/2)/13)/e**6 + 4*B*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**6 + 2*B*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**6
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.33437, 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)^2*(B*x + A)*(e*x + d)^(5/2),x, algorithm="giac")
[Out]
Done