3.1613 \(\int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=71 \[ -\frac{4 b (d+e x)^{5/2} (b d-a e)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 e^3}+\frac{2 b^2 (d+e x)^{7/2}}{7 e^3} \]

[Out]

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

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Rubi [A]  time = 0.083162, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{4 b (d+e x)^{5/2} (b d-a e)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 e^3}+\frac{2 b^2 (d+e x)^{7/2}}{7 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.6573, size = 65, normalized size = 0.92 \[ \frac{2 b^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{3}} + \frac{4 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )}{5 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}}{3 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0676838, size = 61, normalized size = 0.86 \[ \frac{2 (d+e x)^{3/2} \left (35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(3/2)*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x
+ 15*e^2*x^2)))/(105*e^3)

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Maple [A]  time = 0.01, size = 63, normalized size = 0.9 \[{\frac{30\,{x}^{2}{b}^{2}{e}^{2}+84\,xab{e}^{2}-24\,x{b}^{2}de+70\,{a}^{2}{e}^{2}-56\,abde+16\,{b}^{2}{d}^{2}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(e*x+d)^(3/2)*(15*b^2*e^2*x^2+42*a*b*e^2*x-12*b^2*d*e*x+35*a^2*e^2-28*a*b*
d*e+8*b^2*d^2)/e^3

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Maxima [A]  time = 0.724416, size = 92, normalized size = 1.3 \[ \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{2} - 42 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*(e*x + d)^(7/2)*b^2 - 42*(b^2*d - a*b*e)*(e*x + d)^(5/2) + 35*(b^2*d^2
- 2*a*b*d*e + a^2*e^2)*(e*x + d)^(3/2))/e^3

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Fricas [A]  time = 0.205438, size = 134, normalized size = 1.89 \[ \frac{2 \,{\left (15 \, b^{2} e^{3} x^{3} + 8 \, b^{2} d^{3} - 28 \, a b d^{2} e + 35 \, a^{2} d e^{2} + 3 \,{\left (b^{2} d e^{2} + 14 \, a b e^{3}\right )} x^{2} -{\left (4 \, b^{2} d^{2} e - 14 \, a b d e^{2} - 35 \, a^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*b^2*e^3*x^3 + 8*b^2*d^3 - 28*a*b*d^2*e + 35*a^2*d*e^2 + 3*(b^2*d*e^2 +
14*a*b*e^3)*x^2 - (4*b^2*d^2*e - 14*a*b*d*e^2 - 35*a^2*e^3)*x)*sqrt(e*x + d)/e^
3

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Sympy [A]  time = 4.08017, size = 85, normalized size = 1.2 \[ \frac{2 \left (\frac{b^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 a b e - 2 b^{2} d\right )}{5 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{2} e^{2} - 2 a b d e + b^{2} d^{2}\right )}{3 e^{2}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b**2*(d + e*x)**(7/2)/(7*e**2) + (d + e*x)**(5/2)*(2*a*b*e - 2*b**2*d)/(5*e**
2) + (d + e*x)**(3/2)*(a**2*e**2 - 2*a*b*d*e + b**2*d**2)/(3*e**2))/e

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GIAC/XCAS [A]  time = 0.219475, size = 126, normalized size = 1.77 \[ \frac{2}{105} \,{\left (14 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b e^{\left (-1\right )} +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} b^{2} e^{\left (-14\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(14*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*e^(-1) + (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)*b^2*e^(-14)
+ 35*(x*e + d)^(3/2)*a^2)*e^(-1)