3.2113 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=370 \[ \frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^7 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}{e^7 (a+b x)}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^7 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) -
(4*b*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)
) + (6*b^2*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a
+ b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
7*e^7*(a + b*x)) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2])/(3*e^7*(a + b*x)) - (12*b^5*(b*d - a*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^6*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(13*e^7*(a + b*x))

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Rubi [A]  time = 0.404534, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^7 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}{e^7 (a+b x)}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^7 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) -
(4*b*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)
) + (6*b^2*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a
+ b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
7*e^7*(a + b*x)) + (10*b^4*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2])/(3*e^7*(a + b*x)) - (12*b^5*(b*d - a*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^6*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(13*e^7*(a + b*x))

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Rubi in Sympy [A]  time = 62.4383, size = 323, normalized size = 0.87 \[ \frac{2 \left (a + b x\right ) \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{13 e} + \frac{24 \sqrt{d + e x} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{143 e^{2}} + \frac{16 \left (5 a + 5 b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{429 e^{3}} + \frac{640 \sqrt{d + e x} \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3003 e^{4}} + \frac{256 \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3003 e^{5}} + \frac{1024 \sqrt{d + e x} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3003 e^{6}} + \frac{2048 \sqrt{d + e x} \left (a e - b d\right )^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3003 e^{7} \left (a + b x\right )} \]

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]

2*(a + b*x)*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(13*e) + 24*sqrt(d
+ e*x)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(143*e**2) + 16*(5*a + 5
*b*x)*sqrt(d + e*x)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(429*e**3
) + 640*sqrt(d + e*x)*(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3003*e
**4) + 256*(3*a + 3*b*x)*sqrt(d + e*x)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2
*x**2)/(3003*e**5) + 1024*sqrt(d + e*x)*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x + b**
2*x**2)/(3003*e**6) + 2048*sqrt(d + e*x)*(a*e - b*d)**6*sqrt(a**2 + 2*a*b*x + b*
*2*x**2)/(3003*e**7*(a + b*x))

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Mathematica [A]  time = 0.261708, size = 308, normalized size = 0.83 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (3003 a^6 e^6+6006 a^5 b e^5 (e x-2 d)+3003 a^4 b^2 e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+1716 a^3 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 a^2 b^4 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+26 a b^5 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+b^6 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{3003 e^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(3003*a^6*e^6 + 6006*a^5*b*e^5*(-2*d + e*x) +
3003*a^4*b^2*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 1716*a^3*b^3*e^3*(-16*d^3 + 8*
d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 143*a^2*b^4*e^2*(128*d^4 - 64*d^3*e*x + 48*
d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 26*a*b^5*e*(-256*d^5 + 128*d^4*e*x -
96*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) + b^6*(1024*d^6 - 5
12*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5
+ 231*e^6*x^6)))/(3003*e^7*(a + b*x))

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Maple [A]  time = 0.013, size = 393, normalized size = 1.1 \[{\frac{462\,{x}^{6}{b}^{6}{e}^{6}+3276\,{x}^{5}a{b}^{5}{e}^{6}-504\,{x}^{5}{b}^{6}d{e}^{5}+10010\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-3640\,{x}^{4}a{b}^{5}d{e}^{5}+560\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+17160\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-11440\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+4160\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-640\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+18018\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-20592\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+13728\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-4992\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+768\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+12012\,x{a}^{5}b{e}^{6}-24024\,x{a}^{4}{b}^{2}d{e}^{5}+27456\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-18304\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+6656\,xa{b}^{5}{d}^{4}{e}^{2}-1024\,x{b}^{6}{d}^{5}e+6006\,{a}^{6}{e}^{6}-24024\,{a}^{5}bd{e}^{5}+48048\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-54912\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+36608\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-13312\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{3003\, \left ( bx+a \right ) ^{5}{e}^{7}}\sqrt{ex+d} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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]

2/3003*(e*x+d)^(1/2)*(231*b^6*e^6*x^6+1638*a*b^5*e^6*x^5-252*b^6*d*e^5*x^5+5005*
a^2*b^4*e^6*x^4-1820*a*b^5*d*e^5*x^4+280*b^6*d^2*e^4*x^4+8580*a^3*b^3*e^6*x^3-57
20*a^2*b^4*d*e^5*x^3+2080*a*b^5*d^2*e^4*x^3-320*b^6*d^3*e^3*x^3+9009*a^4*b^2*e^6
*x^2-10296*a^3*b^3*d*e^5*x^2+6864*a^2*b^4*d^2*e^4*x^2-2496*a*b^5*d^3*e^3*x^2+384
*b^6*d^4*e^2*x^2+6006*a^5*b*e^6*x-12012*a^4*b^2*d*e^5*x+13728*a^3*b^3*d^2*e^4*x-
9152*a^2*b^4*d^3*e^3*x+3328*a*b^5*d^4*e^2*x-512*b^6*d^5*e*x+3003*a^6*e^6-12012*a
^5*b*d*e^5+24024*a^4*b^2*d^2*e^4-27456*a^3*b^3*d^3*e^3+18304*a^2*b^4*d^4*e^2-665
6*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [A]  time = 0.72027, size = 1023, normalized size = 2.76 \[ \text{result too large to display} \]

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]

2/693*(63*b^5*e^6*x^6 - 256*b^5*d^6 + 1408*a*b^4*d^5*e - 3168*a^2*b^3*d^4*e^2 +
3696*a^3*b^2*d^3*e^3 - 2310*a^4*b*d^2*e^4 + 693*a^5*d*e^5 - 7*(b^5*d*e^5 - 55*a*
b^4*e^6)*x^5 + 5*(2*b^5*d^2*e^4 - 11*a*b^4*d*e^5 + 198*a^2*b^3*e^6)*x^4 - 2*(8*b
^5*d^3*e^3 - 44*a*b^4*d^2*e^4 + 99*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (32*b^
5*d^4*e^2 - 176*a*b^4*d^3*e^3 + 396*a^2*b^3*d^2*e^4 - 462*a^3*b^2*d*e^5 + 1155*a
^4*b*e^6)*x^2 - (128*b^5*d^5*e - 704*a*b^4*d^4*e^2 + 1584*a^2*b^3*d^3*e^3 - 1848
*a^3*b^2*d^2*e^4 + 1155*a^4*b*d*e^5 - 693*a^5*e^6)*x)*a/(sqrt(e*x + d)*e^6) + 2/
9009*(693*b^5*e^7*x^7 + 3072*b^5*d^7 - 16640*a*b^4*d^6*e + 36608*a^2*b^3*d^5*e^2
- 41184*a^3*b^2*d^4*e^3 + 24024*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 - 63*(b^5*d*e^
6 - 65*a*b^4*e^7)*x^6 + 7*(12*b^5*d^2*e^5 - 65*a*b^4*d*e^6 + 1430*a^2*b^3*e^7)*x
^5 - 10*(12*b^5*d^3*e^4 - 65*a*b^4*d^2*e^5 + 143*a^2*b^3*d*e^6 - 1287*a^3*b^2*e^
7)*x^4 + (192*b^5*d^4*e^3 - 1040*a*b^4*d^3*e^4 + 2288*a^2*b^3*d^2*e^5 - 2574*a^3
*b^2*d*e^6 + 9009*a^4*b*e^7)*x^3 - (384*b^5*d^5*e^2 - 2080*a*b^4*d^4*e^3 + 4576*
a^2*b^3*d^3*e^4 - 5148*a^3*b^2*d^2*e^5 + 3003*a^4*b*d*e^6 - 3003*a^5*e^7)*x^2 +
(1536*b^5*d^6*e - 8320*a*b^4*d^5*e^2 + 18304*a^2*b^3*d^4*e^3 - 20592*a^3*b^2*d^3
*e^4 + 12012*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*b/(sqrt(e*x + d)*e^7)

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Fricas [A]  time = 0.321291, size = 481, normalized size = 1.3 \[ \frac{2 \,{\left (231 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 6656 \, a b^{5} d^{5} e + 18304 \, a^{2} b^{4} d^{4} e^{2} - 27456 \, a^{3} b^{3} d^{3} e^{3} + 24024 \, a^{4} b^{2} d^{2} e^{4} - 12012 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} - 126 \,{\left (2 \, b^{6} d e^{5} - 13 \, a b^{5} e^{6}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{2} e^{4} - 52 \, a b^{5} d e^{5} + 143 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (16 \, b^{6} d^{3} e^{3} - 104 \, a b^{5} d^{2} e^{4} + 286 \, a^{2} b^{4} d e^{5} - 429 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{4} e^{2} - 832 \, a b^{5} d^{3} e^{3} + 2288 \, a^{2} b^{4} d^{2} e^{4} - 3432 \, a^{3} b^{3} d e^{5} + 3003 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{5} e - 1664 \, a b^{5} d^{4} e^{2} + 4576 \, a^{2} b^{4} d^{3} e^{3} - 6864 \, a^{3} b^{3} d^{2} e^{4} + 6006 \, a^{4} b^{2} d e^{5} - 3003 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{3003 \, e^{7}} \]

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]

2/3003*(231*b^6*e^6*x^6 + 1024*b^6*d^6 - 6656*a*b^5*d^5*e + 18304*a^2*b^4*d^4*e^
2 - 27456*a^3*b^3*d^3*e^3 + 24024*a^4*b^2*d^2*e^4 - 12012*a^5*b*d*e^5 + 3003*a^6
*e^6 - 126*(2*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*e^4 - 52*a*b^5*d*e^5
+ 143*a^2*b^4*e^6)*x^4 - 20*(16*b^6*d^3*e^3 - 104*a*b^5*d^2*e^4 + 286*a^2*b^4*d
*e^5 - 429*a^3*b^3*e^6)*x^3 + 3*(128*b^6*d^4*e^2 - 832*a*b^5*d^3*e^3 + 2288*a^2*
b^4*d^2*e^4 - 3432*a^3*b^3*d*e^5 + 3003*a^4*b^2*e^6)*x^2 - 2*(256*b^6*d^5*e - 16
64*a*b^5*d^4*e^2 + 4576*a^2*b^4*d^3*e^3 - 6864*a^3*b^3*d^2*e^4 + 6006*a^4*b^2*d*
e^5 - 3003*a^5*b*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 = 0.327385, size = 657, normalized size = 1.78 \[ \frac{2}{3003} \,{\left (6006 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{5} b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a^{4} b^{2} e^{\left (-10\right )}{\rm sign}\left (b x + a\right ) + 1716 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} a^{3} b^{3} e^{\left (-21\right )}{\rm sign}\left (b x + a\right ) + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} a^{2} b^{4} e^{\left (-36\right )}{\rm sign}\left (b x + a\right ) + 26 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} a b^{5} e^{\left (-55\right )}{\rm sign}\left (b x + a\right ) +{\left (231 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{72} - 1638 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{72} + 5005 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{72} - 8580 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{72} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{72} - 6006 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{72} + 3003 \, \sqrt{x e + d} d^{6} e^{72}\right )} b^{6} e^{\left (-78\right )}{\rm sign}\left (b x + a\right ) + 3003 \, \sqrt{x e + d} a^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

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]

2/3003*(6006*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*e^(-1)*sign(b*x + a) +
3003*(3*(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^
8)*a^4*b^2*e^(-10)*sign(b*x + a) + 1716*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(
5/2)*d*e^18 + 35*(x*e + d)^(3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*a^3*b^3*e
^(-21)*sign(b*x + a) + 143*(35*(x*e + d)^(9/2)*e^32 - 180*(x*e + d)^(7/2)*d*e^32
+ 378*(x*e + d)^(5/2)*d^2*e^32 - 420*(x*e + d)^(3/2)*d^3*e^32 + 315*sqrt(x*e +
d)*d^4*e^32)*a^2*b^4*e^(-36)*sign(b*x + a) + 26*(63*(x*e + d)^(11/2)*e^50 - 385*
(x*e + d)^(9/2)*d*e^50 + 990*(x*e + d)^(7/2)*d^2*e^50 - 1386*(x*e + d)^(5/2)*d^3
*e^50 + 1155*(x*e + d)^(3/2)*d^4*e^50 - 693*sqrt(x*e + d)*d^5*e^50)*a*b^5*e^(-55
)*sign(b*x + a) + (231*(x*e + d)^(13/2)*e^72 - 1638*(x*e + d)^(11/2)*d*e^72 + 50
05*(x*e + d)^(9/2)*d^2*e^72 - 8580*(x*e + d)^(7/2)*d^3*e^72 + 9009*(x*e + d)^(5/
2)*d^4*e^72 - 6006*(x*e + d)^(3/2)*d^5*e^72 + 3003*sqrt(x*e + d)*d^6*e^72)*b^6*e
^(-78)*sign(b*x + a) + 3003*sqrt(x*e + d)*a^6*sign(b*x + a))*e^(-1)