3.1792 \(\int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)
Optimal. Leaf size=218 \[ -\frac{2 b^3 (d+e x)^{17/2} (-4 a B e-A b e+5 b B d)}{17 e^6}+\frac{4 b^2 (d+e x)^{15/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{15 e^6}-\frac{4 b (d+e x)^{13/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{13 e^6}+\frac{2 (d+e x)^{11/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{11 e^6}-\frac{2 (d+e x)^{9/2} (b d-a e)^4 (B d-A e)}{9 e^6}+\frac{2 b^4 B (d+e x)^{19/2}}{19 e^6} \]
[Out]
(-2*(b*d - a*e)^4*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^6) + (2*(b*d - a*e)^3*(5*b*B
*d - 4*A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d -
3*A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^6) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2
*A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e
)*(d + e*x)^(17/2))/(17*e^6) + (2*b^4*B*(d + e*x)^(19/2))/(19*e^6)
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Rubi [A] time = 0.357957, 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
\[ -\frac{2 b^3 (d+e x)^{17/2} (-4 a B e-A b e+5 b B d)}{17 e^6}+\frac{4 b^2 (d+e x)^{15/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{15 e^6}-\frac{4 b (d+e x)^{13/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{13 e^6}+\frac{2 (d+e x)^{11/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{11 e^6}-\frac{2 (d+e x)^{9/2} (b d-a e)^4 (B d-A e)}{9 e^6}+\frac{2 b^4 B (d+e x)^{19/2}}{19 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^(7/2)*(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)^(9/2))/(9*e^6) + (2*(b*d - a*e)^3*(5*b*B
*d - 4*A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d -
3*A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^6) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2
*A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e
)*(d + e*x)^(17/2))/(17*e^6) + (2*b^4*B*(d + e*x)^(19/2))/(19*e^6)
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Rubi in Sympy [A] time = 110.91, size = 221, normalized size = 1.01 \[ \frac{2 B b^{4} \left (d + e x\right )^{\frac{19}{2}}}{19 e^{6}} + \frac{2 b^{3} \left (d + e x\right )^{\frac{17}{2}} \left (A b e + 4 B a e - 5 B b d\right )}{17 e^{6}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{15 e^{6}} + \frac{4 b \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{13 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{11 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{9 e^{6}} \]
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)**2,x)
[Out]
2*B*b**4*(d + e*x)**(19/2)/(19*e**6) + 2*b**3*(d + e*x)**(17/2)*(A*b*e + 4*B*a*e
- 5*B*b*d)/(17*e**6) + 4*b**2*(d + e*x)**(15/2)*(a*e - b*d)*(2*A*b*e + 3*B*a*e
- 5*B*b*d)/(15*e**6) + 4*b*(d + e*x)**(13/2)*(a*e - b*d)**2*(3*A*b*e + 2*B*a*e -
5*B*b*d)/(13*e**6) + 2*(d + e*x)**(11/2)*(a*e - b*d)**3*(4*A*b*e + B*a*e - 5*B*
b*d)/(11*e**6) + 2*(d + e*x)**(9/2)*(A*e - B*d)*(a*e - b*d)**4/(9*e**6)
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Mathematica [A] time = 0.796377, size = 339, normalized size = 1.56 \[ \frac{2 (d+e x)^{9/2} \left (20995 a^4 e^4 (11 A e-2 B d+9 B e x)+6460 a^3 b e^3 \left (13 A e (9 e x-2 d)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )-1938 a^2 b^2 e^2 \left (B \left (16 d^3-72 d^2 e x+198 d e^2 x^2-429 e^3 x^3\right )-5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+76 a b^3 e \left (17 A e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+B \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )+b^4 \left (19 A e \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )-5 B \left (256 d^5-1152 d^4 e x+3168 d^3 e^2 x^2-6864 d^2 e^3 x^3+12870 d e^4 x^4-21879 e^5 x^5\right )\right )\right )}{2078505 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(2*(d + e*x)^(9/2)*(20995*a^4*e^4*(-2*B*d + 11*A*e + 9*B*e*x) + 6460*a^3*b*e^3*(
13*A*e*(-2*d + 9*e*x) + B*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) - 1938*a^2*b^2*e^2*(-
5*A*e*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + B*(16*d^3 - 72*d^2*e*x + 198*d*e^2*x^2 -
429*e^3*x^3)) + 76*a*b^3*e*(17*A*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*
e^3*x^3) + B*(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)) + b^4*(19*A*e*(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) - 5*B*(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))))/(2078505*e^6)
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Maple [B] time = 0.014, size = 469, normalized size = 2.2 \[{\frac{218790\,{b}^{4}B{x}^{5}{e}^{5}+244530\,A{b}^{4}{e}^{5}{x}^{4}+978120\,Ba{b}^{3}{e}^{5}{x}^{4}-128700\,B{b}^{4}d{e}^{4}{x}^{4}+1108536\,Aa{b}^{3}{e}^{5}{x}^{3}-130416\,A{b}^{4}d{e}^{4}{x}^{3}+1662804\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-521664\,Ba{b}^{3}d{e}^{4}{x}^{3}+68640\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+1918620\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-511632\,Aa{b}^{3}d{e}^{4}{x}^{2}+60192\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+1279080\,B{a}^{3}b{e}^{5}{x}^{2}-767448\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+240768\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-31680\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+1511640\,A{a}^{3}b{e}^{5}x-697680\,A{a}^{2}{b}^{2}d{e}^{4}x+186048\,Aa{b}^{3}{d}^{2}{e}^{3}x-21888\,A{b}^{4}{d}^{3}{e}^{2}x+377910\,B{a}^{4}{e}^{5}x-465120\,B{a}^{3}bd{e}^{4}x+279072\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-87552\,Ba{b}^{3}{d}^{3}{e}^{2}x+11520\,B{b}^{4}{d}^{4}ex+461890\,A{a}^{4}{e}^{5}-335920\,Ad{a}^{3}b{e}^{4}+155040\,A{a}^{2}{b}^{2}{d}^{2}{e}^{3}-41344\,Aa{b}^{3}{d}^{3}{e}^{2}+4864\,A{d}^{4}{b}^{4}e-83980\,B{a}^{4}d{e}^{4}+103360\,B{d}^{2}{a}^{3}b{e}^{3}-62016\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+19456\,B{d}^{4}a{b}^{3}e-2560\,{b}^{4}B{d}^{5}}{2078505\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]
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)^2,x)
[Out]
2/2078505*(e*x+d)^(9/2)*(109395*B*b^4*e^5*x^5+122265*A*b^4*e^5*x^4+489060*B*a*b^
3*e^5*x^4-64350*B*b^4*d*e^4*x^4+554268*A*a*b^3*e^5*x^3-65208*A*b^4*d*e^4*x^3+831
402*B*a^2*b^2*e^5*x^3-260832*B*a*b^3*d*e^4*x^3+34320*B*b^4*d^2*e^3*x^3+959310*A*
a^2*b^2*e^5*x^2-255816*A*a*b^3*d*e^4*x^2+30096*A*b^4*d^2*e^3*x^2+639540*B*a^3*b*
e^5*x^2-383724*B*a^2*b^2*d*e^4*x^2+120384*B*a*b^3*d^2*e^3*x^2-15840*B*b^4*d^3*e^
2*x^2+755820*A*a^3*b*e^5*x-348840*A*a^2*b^2*d*e^4*x+93024*A*a*b^3*d^2*e^3*x-1094
4*A*b^4*d^3*e^2*x+188955*B*a^4*e^5*x-232560*B*a^3*b*d*e^4*x+139536*B*a^2*b^2*d^2
*e^3*x-43776*B*a*b^3*d^3*e^2*x+5760*B*b^4*d^4*e*x+230945*A*a^4*e^5-167960*A*a^3*
b*d*e^4+77520*A*a^2*b^2*d^2*e^3-20672*A*a*b^3*d^3*e^2+2432*A*b^4*d^4*e-41990*B*a
^4*d*e^4+51680*B*a^3*b*d^2*e^3-31008*B*a^2*b^2*d^3*e^2+9728*B*a*b^3*d^4*e-1280*B
*b^4*d^5)/e^6
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Maxima [A] time = 0.725306, size = 552, normalized size = 2.53 \[ \frac{2 \,{\left (109395 \,{\left (e x + d\right )}^{\frac{19}{2}} B b^{4} - 122265 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 277134 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 319770 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 188955 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 230945 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{2078505 \, e^{6}} \]
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)^(7/2),x, algorithm="maxima")
[Out]
2/2078505*(109395*(e*x + d)^(19/2)*B*b^4 - 122265*(5*B*b^4*d - (4*B*a*b^3 + A*b^
4)*e)*(e*x + d)^(17/2) + 277134*(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)^(15/2) - 319770*(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)^(13/2) + 188955*(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)^(11/2) - 230945*(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)^(9/2))/e^6
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Fricas [A] time = 0.290437, size = 1207, normalized size = 5.54 \[ \text{result too large to display} \]
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)^(7/2),x, algorithm="fricas")
[Out]
2/2078505*(109395*B*b^4*e^9*x^9 - 1280*B*b^4*d^9 + 230945*A*a^4*d^4*e^5 + 2432*(
4*B*a*b^3 + A*b^4)*d^8*e - 10336*(3*B*a^2*b^2 + 2*A*a*b^3)*d^7*e^2 + 25840*(2*B*
a^3*b + 3*A*a^2*b^2)*d^6*e^3 - 41990*(B*a^4 + 4*A*a^3*b)*d^5*e^4 + 6435*(58*B*b^
4*d*e^8 + 19*(4*B*a*b^3 + A*b^4)*e^9)*x^8 + 858*(505*B*b^4*d^2*e^7 + 494*(4*B*a*
b^3 + A*b^4)*d*e^8 + 323*(3*B*a^2*b^2 + 2*A*a*b^3)*e^9)*x^7 + 66*(2620*B*b^4*d^3
*e^6 + 7619*(4*B*a*b^3 + A*b^4)*d^2*e^7 + 14858*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^8
+ 4845*(2*B*a^3*b + 3*A*a^2*b^2)*e^9)*x^6 + 9*(35*B*b^4*d^4*e^5 + 23028*(4*B*a*b
^3 + A*b^4)*d^3*e^6 + 133076*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^7 + 129200*(2*B*a^3
*b + 3*A*a^2*b^2)*d*e^8 + 20995*(B*a^4 + 4*A*a^3*b)*e^9)*x^5 - 5*(70*B*b^4*d^5*e
^4 - 46189*A*a^4*e^9 - 133*(4*B*a*b^3 + A*b^4)*d^4*e^5 - 103360*(3*B*a^2*b^2 + 2
*A*a*b^3)*d^3*e^6 - 295868*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^7 - 142766*(B*a^4 + 4
*A*a^3*b)*d*e^8)*x^4 + 10*(40*B*b^4*d^6*e^3 + 92378*A*a^4*d*e^8 - 76*(4*B*a*b^3
+ A*b^4)*d^5*e^4 + 323*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^5 + 68476*(2*B*a^3*b + 3*
A*a^2*b^2)*d^3*e^6 + 96577*(B*a^4 + 4*A*a^3*b)*d^2*e^7)*x^3 - 6*(80*B*b^4*d^7*e^
2 - 230945*A*a^4*d^2*e^7 - 152*(4*B*a*b^3 + A*b^4)*d^6*e^3 + 646*(3*B*a^2*b^2 +
2*A*a*b^3)*d^5*e^4 - 1615*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^5 - 83980*(B*a^4 + 4*A
*a^3*b)*d^3*e^6)*x^2 + (640*B*b^4*d^8*e + 923780*A*a^4*d^3*e^6 - 1216*(4*B*a*b^3
+ A*b^4)*d^7*e^2 + 5168*(3*B*a^2*b^2 + 2*A*a*b^3)*d^6*e^3 - 12920*(2*B*a^3*b +
3*A*a^2*b^2)*d^5*e^4 + 20995*(B*a^4 + 4*A*a^3*b)*d^4*e^5)*x)*sqrt(e*x + d)/e^6
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Sympy [A] time = 57.4009, size = 2091, normalized size = 9.59 \[ \text{result too large to display} \]
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)**2,x)
[Out]
Piecewise((2*A*a**4*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**4*d**3*x*sqrt(d + e*x)/9 +
4*A*a**4*d**2*e*x**2*sqrt(d + e*x)/3 + 8*A*a**4*d*e**2*x**3*sqrt(d + e*x)/9 + 2
*A*a**4*e**3*x**4*sqrt(d + e*x)/9 - 16*A*a**3*b*d**5*sqrt(d + e*x)/(99*e**2) + 8
*A*a**3*b*d**4*x*sqrt(d + e*x)/(99*e) + 64*A*a**3*b*d**3*x**2*sqrt(d + e*x)/33 +
368*A*a**3*b*d**2*e*x**3*sqrt(d + e*x)/99 + 272*A*a**3*b*d*e**2*x**4*sqrt(d + e
*x)/99 + 8*A*a**3*b*e**3*x**5*sqrt(d + e*x)/11 + 32*A*a**2*b**2*d**6*sqrt(d + e*
x)/(429*e**3) - 16*A*a**2*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 4*A*a**2*b**2*d
**4*x**2*sqrt(d + e*x)/(143*e) + 848*A*a**2*b**2*d**3*x**3*sqrt(d + e*x)/429 + 1
832*A*a**2*b**2*d**2*e*x**4*sqrt(d + e*x)/429 + 480*A*a**2*b**2*d*e**2*x**5*sqrt
(d + e*x)/143 + 12*A*a**2*b**2*e**3*x**6*sqrt(d + e*x)/13 - 128*A*a*b**3*d**7*sq
rt(d + e*x)/(6435*e**4) + 64*A*a*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 16*A*a*
b**3*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 8*A*a*b**3*d**4*x**3*sqrt(d + e*x)/(1
287*e) + 1280*A*a*b**3*d**3*x**4*sqrt(d + e*x)/1287 + 1648*A*a*b**3*d**2*e*x**5*
sqrt(d + e*x)/715 + 368*A*a*b**3*d*e**2*x**6*sqrt(d + e*x)/195 + 8*A*a*b**3*e**3
*x**7*sqrt(d + e*x)/15 + 256*A*b**4*d**8*sqrt(d + e*x)/(109395*e**5) - 128*A*b**
4*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*A*b**4*d**6*x**2*sqrt(d + e*x)/(36465*
e**3) - 16*A*b**4*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*A*b**4*d**4*x**4*sqr
t(d + e*x)/(21879*e) + 2424*A*b**4*d**3*x**5*sqrt(d + e*x)/12155 + 1604*A*b**4*d
**2*e*x**6*sqrt(d + e*x)/3315 + 104*A*b**4*d*e**2*x**7*sqrt(d + e*x)/255 + 2*A*b
**4*e**3*x**8*sqrt(d + e*x)/17 - 4*B*a**4*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*a**
4*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**4*d**3*x**2*sqrt(d + e*x)/33 + 92*B*a**4
*d**2*e*x**3*sqrt(d + e*x)/99 + 68*B*a**4*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**
4*e**3*x**5*sqrt(d + e*x)/11 + 64*B*a**3*b*d**6*sqrt(d + e*x)/(1287*e**3) - 32*B
*a**3*b*d**5*x*sqrt(d + e*x)/(1287*e**2) + 8*B*a**3*b*d**4*x**2*sqrt(d + e*x)/(4
29*e) + 1696*B*a**3*b*d**3*x**3*sqrt(d + e*x)/1287 + 3664*B*a**3*b*d**2*e*x**4*s
qrt(d + e*x)/1287 + 320*B*a**3*b*d*e**2*x**5*sqrt(d + e*x)/143 + 8*B*a**3*b*e**3
*x**6*sqrt(d + e*x)/13 - 64*B*a**2*b**2*d**7*sqrt(d + e*x)/(2145*e**4) + 32*B*a*
*2*b**2*d**6*x*sqrt(d + e*x)/(2145*e**3) - 8*B*a**2*b**2*d**5*x**2*sqrt(d + e*x)
/(715*e**2) + 4*B*a**2*b**2*d**4*x**3*sqrt(d + e*x)/(429*e) + 640*B*a**2*b**2*d*
*3*x**4*sqrt(d + e*x)/429 + 2472*B*a**2*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 184
*B*a**2*b**2*d*e**2*x**6*sqrt(d + e*x)/65 + 4*B*a**2*b**2*e**3*x**7*sqrt(d + e*x
)/5 + 1024*B*a*b**3*d**8*sqrt(d + e*x)/(109395*e**5) - 512*B*a*b**3*d**7*x*sqrt(
d + e*x)/(109395*e**4) + 128*B*a*b**3*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 64*
B*a*b**3*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 56*B*a*b**3*d**4*x**4*sqrt(d + e
*x)/(21879*e) + 9696*B*a*b**3*d**3*x**5*sqrt(d + e*x)/12155 + 6416*B*a*b**3*d**2
*e*x**6*sqrt(d + e*x)/3315 + 416*B*a*b**3*d*e**2*x**7*sqrt(d + e*x)/255 + 8*B*a*
b**3*e**3*x**8*sqrt(d + e*x)/17 - 512*B*b**4*d**9*sqrt(d + e*x)/(415701*e**6) +
256*B*b**4*d**8*x*sqrt(d + e*x)/(415701*e**5) - 64*B*b**4*d**7*x**2*sqrt(d + e*x
)/(138567*e**4) + 160*B*b**4*d**6*x**3*sqrt(d + e*x)/(415701*e**3) - 140*B*b**4*
d**5*x**4*sqrt(d + e*x)/(415701*e**2) + 14*B*b**4*d**4*x**5*sqrt(d + e*x)/(46189
*e) + 2096*B*b**4*d**3*x**6*sqrt(d + e*x)/12597 + 404*B*b**4*d**2*e*x**7*sqrt(d
+ e*x)/969 + 116*B*b**4*d*e**2*x**8*sqrt(d + e*x)/323 + 2*B*b**4*e**3*x**9*sqrt(
d + e*x)/19, Ne(e, 0)), (d**(7/2)*(A*a**4*x + 2*A*a**3*b*x**2 + 2*A*a**2*b**2*x*
*3 + A*a*b**3*x**4 + A*b**4*x**5/5 + B*a**4*x**2/2 + 4*B*a**3*b*x**3/3 + 3*B*a**
2*b**2*x**4/2 + 4*B*a*b**3*x**5/5 + B*b**4*x**6/6), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.351192, 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)^(7/2),x, algorithm="giac")
[Out]
Done