3.2054 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=154 \[ -\frac{10 b^4 (d+e x)^{9/2} (b d-a e)}{9 e^6}+\frac{20 b^3 (d+e x)^{7/2} (b d-a e)^2}{7 e^6}-\frac{4 b^2 (d+e x)^{5/2} (b d-a e)^3}{e^6}+\frac{10 b (d+e x)^{3/2} (b d-a e)^4}{3 e^6}-\frac{2 \sqrt{d+e x} (b d-a e)^5}{e^6}+\frac{2 b^5 (d+e x)^{11/2}}{11 e^6} \]

[Out]

(-2*(b*d - a*e)^5*Sqrt[d + e*x])/e^6 + (10*b*(b*d - a*e)^4*(d + e*x)^(3/2))/(3*e
^6) - (4*b^2*(b*d - a*e)^3*(d + e*x)^(5/2))/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x
)^(7/2))/(7*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^6) + (2*b^5*(d + e*
x)^(11/2))/(11*e^6)

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Rubi [A]  time = 0.135066, antiderivative size = 154, 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 \[ -\frac{10 b^4 (d+e x)^{9/2} (b d-a e)}{9 e^6}+\frac{20 b^3 (d+e x)^{7/2} (b d-a e)^2}{7 e^6}-\frac{4 b^2 (d+e x)^{5/2} (b d-a e)^3}{e^6}+\frac{10 b (d+e x)^{3/2} (b d-a e)^4}{3 e^6}-\frac{2 \sqrt{d+e x} (b d-a e)^5}{e^6}+\frac{2 b^5 (d+e x)^{11/2}}{11 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^5*Sqrt[d + e*x])/e^6 + (10*b*(b*d - a*e)^4*(d + e*x)^(3/2))/(3*e
^6) - (4*b^2*(b*d - a*e)^3*(d + e*x)^(5/2))/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x
)^(7/2))/(7*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^6) + (2*b^5*(d + e*
x)^(11/2))/(11*e^6)

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Rubi in Sympy [A]  time = 71.5949, size = 143, normalized size = 0.93 \[ \frac{2 b^{5} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{6}} + \frac{10 b^{4} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )}{9 e^{6}} + \frac{20 b^{3} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}}{7 e^{6}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}}{e^{6}} + \frac{10 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}}{3 e^{6}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{5}}{e^{6}} \]

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)**2/(e*x+d)**(1/2),x)

[Out]

2*b**5*(d + e*x)**(11/2)/(11*e**6) + 10*b**4*(d + e*x)**(9/2)*(a*e - b*d)/(9*e**
6) + 20*b**3*(d + e*x)**(7/2)*(a*e - b*d)**2/(7*e**6) + 4*b**2*(d + e*x)**(5/2)*
(a*e - b*d)**3/e**6 + 10*b*(d + e*x)**(3/2)*(a*e - b*d)**4/(3*e**6) + 2*sqrt(d +
e*x)*(a*e - b*d)**5/e**6

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Mathematica [A]  time = 0.232576, size = 216, normalized size = 1.4 \[ \frac{2 \sqrt{d+e x} \left (693 a^5 e^5+1155 a^4 b e^4 (e x-2 d)+462 a^3 b^2 e^3 \left (8 d^2-4 d e x+3 e^2 x^2\right )+198 a^2 b^3 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+11 a b^4 e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )\right )}{693 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(693*a^5*e^5 + 1155*a^4*b*e^4*(-2*d + e*x) + 462*a^3*b^2*e^3*(8
*d^2 - 4*d*e*x + 3*e^2*x^2) + 198*a^2*b^3*e^2*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2
+ 5*e^3*x^3) + 11*a*b^4*e*(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) + b^5*(-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)))/(693*e^6)

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Maple [B]  time = 0.011, size = 273, normalized size = 1.8 \[{\frac{126\,{x}^{5}{b}^{5}{e}^{5}+770\,{x}^{4}a{b}^{4}{e}^{5}-140\,{x}^{4}{b}^{5}d{e}^{4}+1980\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-880\,{x}^{3}a{b}^{4}d{e}^{4}+160\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+2772\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-2376\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+1056\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-192\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+2310\,x{a}^{4}b{e}^{5}-3696\,x{a}^{3}{b}^{2}d{e}^{4}+3168\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-1408\,xa{b}^{4}{d}^{3}{e}^{2}+256\,x{b}^{5}{d}^{4}e+1386\,{a}^{5}{e}^{5}-4620\,{a}^{4}bd{e}^{4}+7392\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-6336\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+2816\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{693\,{e}^{6}}\sqrt{ex+d}} \]

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)^2/(e*x+d)^(1/2),x)

[Out]

2/693*(63*b^5*e^5*x^5+385*a*b^4*e^5*x^4-70*b^5*d*e^4*x^4+990*a^2*b^3*e^5*x^3-440
*a*b^4*d*e^4*x^3+80*b^5*d^2*e^3*x^3+1386*a^3*b^2*e^5*x^2-1188*a^2*b^3*d*e^4*x^2+
528*a*b^4*d^2*e^3*x^2-96*b^5*d^3*e^2*x^2+1155*a^4*b*e^5*x-1848*a^3*b^2*d*e^4*x+1
584*a^2*b^3*d^2*e^3*x-704*a*b^4*d^3*e^2*x+128*b^5*d^4*e*x+693*a^5*e^5-2310*a^4*b
*d*e^4+3696*a^3*b^2*d^2*e^3-3168*a^2*b^3*d^3*e^2+1408*a*b^4*d^4*e-256*b^5*d^5)*(
e*x+d)^(1/2)/e^6

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Maxima [A]  time = 0.715718, size = 350, normalized size = 2.27 \[ \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{5} - 385 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 990 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 1386 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 693 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{e x + d}\right )}}{693 \, e^{6}} \]

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)/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/693*(63*(e*x + d)^(11/2)*b^5 - 385*(b^5*d - a*b^4*e)*(e*x + d)^(9/2) + 990*(b^
5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(7/2) - 1386*(b^5*d^3 - 3*a*b^4*d^2
*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(5/2) + 1155*(b^5*d^4 - 4*a*b^4*d^
3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(3/2) - 693*(b^
5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4
- a^5*e^5)*sqrt(e*x + d))/e^6

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Fricas [A]  time = 0.287869, size = 352, normalized size = 2.29 \[ \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 1408 \, a b^{4} d^{4} e - 3168 \, a^{2} b^{3} d^{3} e^{2} + 3696 \, a^{3} b^{2} d^{2} e^{3} - 2310 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} - 35 \,{\left (2 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 44 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 88 \, a b^{4} d^{2} e^{3} + 198 \, a^{2} b^{3} d e^{4} - 231 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 704 \, a b^{4} d^{3} e^{2} + 1584 \, a^{2} b^{3} d^{2} e^{3} - 1848 \, a^{3} b^{2} d e^{4} + 1155 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{6}} \]

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)/sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/693*(63*b^5*e^5*x^5 - 256*b^5*d^5 + 1408*a*b^4*d^4*e - 3168*a^2*b^3*d^3*e^2 +
3696*a^3*b^2*d^2*e^3 - 2310*a^4*b*d*e^4 + 693*a^5*e^5 - 35*(2*b^5*d*e^4 - 11*a*b
^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 44*a*b^4*d*e^4 + 99*a^2*b^3*e^5)*x^3 - 6*(16*b
^5*d^3*e^2 - 88*a*b^4*d^2*e^3 + 198*a^2*b^3*d*e^4 - 231*a^3*b^2*e^5)*x^2 + (128*
b^5*d^4*e - 704*a*b^4*d^3*e^2 + 1584*a^2*b^3*d^2*e^3 - 1848*a^3*b^2*d*e^4 + 1155
*a^4*b*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 56.2406, size = 740, normalized size = 4.81 \[ \text{result too large to display} \]

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)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*a**5*d/sqrt(d + e*x) + 2*a**5*(-d/sqrt(d + e*x) - sqrt(d + e*x))
+ 10*a**4*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 10*a**4*b*(d**2/sqrt(d + e*
x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 20*a**3*b**2*d*(d**2/sqrt(d + e
*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 20*a**3*b**2*(-d**3/sqrt(d
+ e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 +
20*a**2*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2)
- (d + e*x)**(5/2)/5)/e**3 + 20*a**2*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d +
e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e*
*3 + 10*a*b**4*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**
(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 10*a*b**4*(-d**5/sqr
t(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x
)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 + 2*b**5*d*(-d**5/s
qrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e
*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 + 2*b**5*(d**6/sq
rt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)*
*(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)
/11)/e**5)/e, Ne(e, 0)), (Piecewise((a**5*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*
x**2)**3/(6*b), True))/sqrt(d), True))

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GIAC/XCAS [A]  time = 0.283656, size = 451, normalized size = 2.93 \[ \frac{2}{693} \,{\left (1155 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{4} b e^{\left (-1\right )} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a^{3} b^{2} e^{\left (-10\right )} + 198 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} a^{2} b^{3} e^{\left (-21\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} a b^{4} e^{\left (-36\right )} +{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} b^{5} e^{\left (-55\right )} + 693 \, \sqrt{x e + d} a^{5}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*b*e^(-1) + 462*(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^3*b^2*e^(-1
0) + 198*(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^2*b^3*e^(-21) + 11*(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*b^4*e^(-36) + (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^5
0)*b^5*e^(-55) + 693*sqrt(x*e + d)*a^5)*e^(-1)