3.1854 \(\int (A+B x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=452 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e
^7*(a + b*x)) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (10*b*(b*d - a*e)^3*(3*b*B*d - 2
*A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x))
- (20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) + (10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a
*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (2*b^
4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(1
3*e^7*(a + b*x)) + (2*b^5*B*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*
e^7*(a + b*x))

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Rubi [A]  time = 0.703269, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e
^7*(a + b*x)) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (10*b*(b*d - a*e)^3*(3*b*B*d - 2
*A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x))
- (20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) + (10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a
*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (2*b^
4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(1
3*e^7*(a + b*x)) + (2*b^5*B*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*
e^7*(a + b*x))

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Rubi in Sympy [A]  time = 74.3825, size = 442, normalized size = 0.98 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{15 b e} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (5 A b e - B a e - 4 B b d\right )}{65 b e^{2}} + \frac{4 \left (5 a + 5 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (5 A b e - B a e - 4 B b d\right )}{715 b e^{3}} + \frac{32 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (5 A b e - B a e - 4 B b d\right )}{1287 b e^{4}} + \frac{64 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - B a e - 4 B b d\right )}{9009 b e^{5}} + \frac{256 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - B a e - 4 B b d\right )}{15015 b e^{6}} + \frac{512 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - B a e - 4 B b d\right )}{45045 b e^{7} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**(3/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(15*b*e) +
2*(d + e*x)**(3/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(5*A*b*e - B*a*e - 4*B*b*
d)/(65*b*e**2) + 4*(5*a + 5*b*x)*(d + e*x)**(3/2)*(a*e - b*d)*(a**2 + 2*a*b*x +
b**2*x**2)**(3/2)*(5*A*b*e - B*a*e - 4*B*b*d)/(715*b*e**3) + 32*(d + e*x)**(3/2)
*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(5*A*b*e - B*a*e - 4*B*b*d)/
(1287*b*e**4) + 64*(3*a + 3*b*x)*(d + e*x)**(3/2)*(a*e - b*d)**3*sqrt(a**2 + 2*a
*b*x + b**2*x**2)*(5*A*b*e - B*a*e - 4*B*b*d)/(9009*b*e**5) + 256*(d + e*x)**(3/
2)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(5*A*b*e - B*a*e - 4*B*b*d)/(
15015*b*e**6) + 512*(d + e*x)**(3/2)*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x + b**2*x
**2)*(5*A*b*e - B*a*e - 4*B*b*d)/(45045*b*e**7*(a + b*x))

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Mathematica [A]  time = 0.786104, size = 490, normalized size = 1.08 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (3003 a^5 e^5 (5 A e-2 B d+3 B e x)+2145 a^4 b e^4 \left (7 A e (3 e x-2 d)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-1430 a^3 b^2 e^3 \left (B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+130 a^2 b^3 e^2 \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-5 a b^4 e \left (5 B \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )-13 A e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+b^5 \left (5 A e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+B \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )\right )}{45045 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(3003*a^5*e^5*(-2*B*d + 5*A*e + 3*B*e*x) +
2145*a^4*b*e^4*(7*A*e*(-2*d + 3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 1430
*a^3*b^2*e^3*(-3*A*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*(16*d^3 - 24*d^2*e*x +
30*d*e^2*x^2 - 35*e^3*x^3)) + 130*a^2*b^3*e^2*(11*A*e*(-16*d^3 + 24*d^2*e*x - 30
*d*e^2*x^2 + 35*e^3*x^3) + B*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^
3*x^3 + 315*e^4*x^4)) - 5*a*b^4*e*(-13*A*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*
x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 5*B*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^
2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^4 - 693*e^5*x^5)) + b^5*(5*A*e*(-256*d^5 + 384
*d^4*e*x - 480*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5) + B*
(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^
4 - 2772*d*e^5*x^5 + 3003*e^6*x^6))))/(45045*e^7*(a + b*x))

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Maple [A]  time = 0.015, size = 689, normalized size = 1.5 \[{\frac{6006\,B{x}^{6}{b}^{5}{e}^{6}+6930\,A{x}^{5}{b}^{5}{e}^{6}+34650\,B{x}^{5}a{b}^{4}{e}^{6}-5544\,B{x}^{5}{b}^{5}d{e}^{5}+40950\,A{x}^{4}a{b}^{4}{e}^{6}-6300\,A{x}^{4}{b}^{5}d{e}^{5}+81900\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-31500\,B{x}^{4}a{b}^{4}d{e}^{5}+5040\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+100100\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}-36400\,A{x}^{3}a{b}^{4}d{e}^{5}+5600\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+100100\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-72800\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+28000\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-4480\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+128700\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-85800\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+31200\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-4800\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+64350\,B{x}^{2}{a}^{4}b{e}^{6}-85800\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+62400\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-24000\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+3840\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+90090\,Ax{a}^{4}b{e}^{6}-102960\,Ax{a}^{3}{b}^{2}d{e}^{5}+68640\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-24960\,Axa{b}^{4}{d}^{3}{e}^{3}+3840\,Ax{b}^{5}{d}^{4}{e}^{2}+18018\,Bx{a}^{5}{e}^{6}-51480\,Bx{a}^{4}bd{e}^{5}+68640\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-49920\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+19200\,Bxa{b}^{4}{d}^{4}{e}^{2}-3072\,Bx{b}^{5}{d}^{5}e+30030\,A{a}^{5}{e}^{6}-60060\,Ad{e}^{5}{a}^{4}b+68640\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-45760\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+16640\,Aa{b}^{4}{d}^{4}{e}^{2}-2560\,A{b}^{5}{d}^{5}e-12012\,Bd{e}^{5}{a}^{5}+34320\,B{a}^{4}b{d}^{2}{e}^{4}-45760\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+33280\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-12800\,Ba{b}^{4}{d}^{5}e+2048\,B{b}^{5}{d}^{6}}{45045\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*B*b^5*e^6*x^6+3465*A*b^5*e^6*x^5+17325*B*a*b^4*e^6*x
^5-2772*B*b^5*d*e^5*x^5+20475*A*a*b^4*e^6*x^4-3150*A*b^5*d*e^5*x^4+40950*B*a^2*b
^3*e^6*x^4-15750*B*a*b^4*d*e^5*x^4+2520*B*b^5*d^2*e^4*x^4+50050*A*a^2*b^3*e^6*x^
3-18200*A*a*b^4*d*e^5*x^3+2800*A*b^5*d^2*e^4*x^3+50050*B*a^3*b^2*e^6*x^3-36400*B
*a^2*b^3*d*e^5*x^3+14000*B*a*b^4*d^2*e^4*x^3-2240*B*b^5*d^3*e^3*x^3+64350*A*a^3*
b^2*e^6*x^2-42900*A*a^2*b^3*d*e^5*x^2+15600*A*a*b^4*d^2*e^4*x^2-2400*A*b^5*d^3*e
^3*x^2+32175*B*a^4*b*e^6*x^2-42900*B*a^3*b^2*d*e^5*x^2+31200*B*a^2*b^3*d^2*e^4*x
^2-12000*B*a*b^4*d^3*e^3*x^2+1920*B*b^5*d^4*e^2*x^2+45045*A*a^4*b*e^6*x-51480*A*
a^3*b^2*d*e^5*x+34320*A*a^2*b^3*d^2*e^4*x-12480*A*a*b^4*d^3*e^3*x+1920*A*b^5*d^4
*e^2*x+9009*B*a^5*e^6*x-25740*B*a^4*b*d*e^5*x+34320*B*a^3*b^2*d^2*e^4*x-24960*B*
a^2*b^3*d^3*e^3*x+9600*B*a*b^4*d^4*e^2*x-1536*B*b^5*d^5*e*x+15015*A*a^5*e^6-3003
0*A*a^4*b*d*e^5+34320*A*a^3*b^2*d^2*e^4-22880*A*a^2*b^3*d^3*e^3+8320*A*a*b^4*d^4
*e^2-1280*A*b^5*d^5*e-6006*B*a^5*d*e^5+17160*B*a^4*b*d^2*e^4-22880*B*a^3*b^2*d^3
*e^3+16640*B*a^2*b^3*d^4*e^2-6400*B*a*b^4*d^5*e+1024*B*b^5*d^6)*((b*x+a)^2)^(5/2
)/e^7/(b*x+a)^5

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Maxima [A]  time = 0.739311, size = 1026, normalized size = 2.27 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2
+ 6864*a^3*b^2*d^3*e^3 - 6006*a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 6
5*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*a^2*b^3*e^6)*x^4 + 1
0*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3
- 3*(32*b^5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^
5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e - 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*
e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(e*x + d)*A
/e^6 + 2/45045*(3003*b^5*e^7*x^7 + 1024*b^5*d^7 - 6400*a*b^4*d^6*e + 16640*a^2*b
^3*d^5*e^2 - 22880*a^3*b^2*d^4*e^3 + 17160*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 + 23
1*(b^5*d*e^6 + 75*a*b^4*e^7)*x^6 - 63*(4*b^5*d^2*e^5 - 25*a*b^4*d*e^6 - 650*a^2*
b^3*e^7)*x^5 + 70*(4*b^5*d^3*e^4 - 25*a*b^4*d^2*e^5 + 65*a^2*b^3*d*e^6 + 715*a^3
*b^2*e^7)*x^4 - 5*(64*b^5*d^4*e^3 - 400*a*b^4*d^3*e^4 + 1040*a^2*b^3*d^2*e^5 - 1
430*a^3*b^2*d*e^6 - 6435*a^4*b*e^7)*x^3 + 3*(128*b^5*d^5*e^2 - 800*a*b^4*d^4*e^3
+ 2080*a^2*b^3*d^3*e^4 - 2860*a^3*b^2*d^2*e^5 + 2145*a^4*b*d*e^6 + 3003*a^5*e^7
)*x^2 - (512*b^5*d^6*e - 3200*a*b^4*d^5*e^2 + 8320*a^2*b^3*d^4*e^3 - 11440*a^3*b
^2*d^3*e^4 + 8580*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*sqrt(e*x + d)*B/e^7

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Fricas [A]  time = 0.286041, size = 948, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^5*e^7*x^7 + 1024*B*b^5*d^7 + 15015*A*a^5*d*e^6 - 1280*(5*B*a*b
^4 + A*b^5)*d^6*e + 8320*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^2 - 22880*(B*a^3*b^2 + A*
a^2*b^3)*d^4*e^3 + 17160*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 6006*(B*a^5 + 5*A*a^4
*b)*d^2*e^5 + 231*(B*b^5*d*e^6 + 15*(5*B*a*b^4 + A*b^5)*e^7)*x^6 - 63*(4*B*b^5*d
^2*e^5 - 5*(5*B*a*b^4 + A*b^5)*d*e^6 - 325*(2*B*a^2*b^3 + A*a*b^4)*e^7)*x^5 + 35
*(8*B*b^5*d^3*e^4 - 10*(5*B*a*b^4 + A*b^5)*d^2*e^5 + 65*(2*B*a^2*b^3 + A*a*b^4)*
d*e^6 + 1430*(B*a^3*b^2 + A*a^2*b^3)*e^7)*x^4 - 5*(64*B*b^5*d^4*e^3 - 80*(5*B*a*
b^4 + A*b^5)*d^3*e^4 + 520*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^5 - 1430*(B*a^3*b^2 + A
*a^2*b^3)*d*e^6 - 6435*(B*a^4*b + 2*A*a^3*b^2)*e^7)*x^3 + 3*(128*B*b^5*d^5*e^2 -
160*(5*B*a*b^4 + A*b^5)*d^4*e^3 + 1040*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^4 - 2860*(
B*a^3*b^2 + A*a^2*b^3)*d^2*e^5 + 2145*(B*a^4*b + 2*A*a^3*b^2)*d*e^6 + 3003*(B*a^
5 + 5*A*a^4*b)*e^7)*x^2 - (512*B*b^5*d^6*e - 15015*A*a^5*e^7 - 640*(5*B*a*b^4 +
A*b^5)*d^5*e^2 + 4160*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^3 - 11440*(B*a^3*b^2 + A*a^2
*b^3)*d^3*e^4 + 8580*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^5 - 3003*(B*a^5 + 5*A*a^4*b)*
d*e^6)*x)*sqrt(e*x + d)/e^7

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.313322, 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)^(5/2)*(B*x + A)*sqrt(e*x + d),x, algorithm="giac")

[Out]

Done