3.2044 \(\int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 b^2 (d+e x)^{9/2} (b d-a e)}{3 e^4}+\frac{6 b (d+e x)^{7/2} (b d-a e)^2}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^3}{5 e^4}+\frac{2 b^3 (d+e x)^{11/2}}{11 e^4} \]

[Out]

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

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Rubi [A]  time = 0.0855903, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 b^2 (d+e x)^{9/2} (b d-a e)}{3 e^4}+\frac{6 b (d+e x)^{7/2} (b d-a e)^2}{7 e^4}-\frac{2 (d+e x)^{5/2} (b d-a e)^3}{5 e^4}+\frac{2 b^3 (d+e x)^{11/2}}{11 e^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 44.311, size = 92, normalized size = 0.92 \[ \frac{2 b^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{2 b^{2} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )}{3 e^{4}} + \frac{6 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}}{7 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}}{5 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.141245, size = 102, normalized size = 1.02 \[ \frac{2 (d+e x)^{5/2} \left (231 a^3 e^3+99 a^2 b e^2 (5 e x-2 d)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(5/2)*(231*a^3*e^3 + 99*a^2*b*e^2*(-2*d + 5*e*x) + 11*a*b^2*e*(8*d^
2 - 20*d*e*x + 35*e^2*x^2) + b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*
x^3)))/(1155*e^4)

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Maple [A]  time = 0.011, size = 116, normalized size = 1.2 \[{\frac{210\,{x}^{3}{b}^{3}{e}^{3}+770\,{x}^{2}a{b}^{2}{e}^{3}-140\,{x}^{2}{b}^{3}d{e}^{2}+990\,x{a}^{2}b{e}^{3}-440\,xa{b}^{2}d{e}^{2}+80\,x{b}^{3}{d}^{2}e+462\,{a}^{3}{e}^{3}-396\,{a}^{2}bd{e}^{2}+176\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{1155\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

2/1155*(e*x+d)^(5/2)*(105*b^3*e^3*x^3+385*a*b^2*e^3*x^2-70*b^3*d*e^2*x^2+495*a^2
*b*e^3*x-220*a*b^2*d*e^2*x+40*b^3*d^2*e*x+231*a^3*e^3-198*a^2*b*d*e^2+88*a*b^2*d
^2*e-16*b^3*d^3)/e^4

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Maxima [A]  time = 0.732725, size = 159, normalized size = 1.59 \[ \frac{2 \,{\left (105 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{3} - 385 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 231 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{1155 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/1155*(105*(e*x + d)^(11/2)*b^3 - 385*(b^3*d - a*b^2*e)*(e*x + d)^(9/2) + 495*(
b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*(e*x + d)^(7/2) - 231*(b^3*d^3 - 3*a*b^2*d^2*
e + 3*a^2*b*d*e^2 - a^3*e^3)*(e*x + d)^(5/2))/e^4

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Fricas [A]  time = 0.285938, size = 292, normalized size = 2.92 \[ \frac{2 \,{\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \,{\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \,{\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d}}{1155 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*
a^3*d^2*e^3 + 35*(4*b^3*d*e^4 + 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d
*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e^3 - 264*a^2*b*d*e^4
- 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 99*a^2*b*d^2*e^3 + 462*a^
3*d*e^4)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 8.61381, size = 386, normalized size = 3.86 \[ a^{3} d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{3} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{6 a^{2} b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{6 a^{2} b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{6 a b^{2} d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{6 a b^{2} \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 b^{3} d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{2 b^{3} \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

a**3*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a*
*3*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 6*a**2*b*d*(-d*(d + e*x)**(3
/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 6*a**2*b*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d +
e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*a*b**2*d*(d**2*(d + e*x)**(3/2)/3 -
2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 6*a*b**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**3 + 2*b**3*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**4 + 2*b**3*(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

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GIAC/XCAS [A]  time = 0.3115, size = 514, normalized size = 5.14 \[ \frac{2}{3465} \,{\left (693 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2} b d e^{\left (-1\right )} + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a b^{2} d e^{\left (-14\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b^{3} d e^{\left (-27\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} d + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a^{2} b e^{\left (-13\right )} + 33 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} a b^{2} e^{\left (-26\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} b^{3} e^{\left (-43\right )} + 231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{3}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

2/3465*(693*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b*d*e^(-1) + 99*(15*(x
*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*a*
b^2*d*e^(-14) + 11*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189*(
x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*b^3*d*e^(-27) + 1155*(x*
e + d)^(3/2)*a^3*d + 99*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 3
5*(x*e + d)^(3/2)*d^2*e^12)*a^2*b*e^(-13) + 33*(35*(x*e + d)^(9/2)*e^24 - 135*(x
*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^
24)*a*b^2*e^(-26) + (315*(x*e + d)^(11/2)*e^40 - 1540*(x*e + d)^(9/2)*d*e^40 + 2
970*(x*e + d)^(7/2)*d^2*e^40 - 2772*(x*e + d)^(5/2)*d^3*e^40 + 1155*(x*e + d)^(3
/2)*d^4*e^40)*b^3*e^(-43) + 231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3)*e
^(-1)