3.2046 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{6 b^2 (d+e x)^{5/2} (b d-a e)}{5 e^4}+\frac{2 b (d+e x)^{3/2} (b d-a e)^2}{e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^3}{e^4}+\frac{2 b^3 (d+e x)^{7/2}}{7 e^4} \]

[Out]

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

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Rubi [A]  time = 0.0830708, antiderivative size = 96, 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 \[ -\frac{6 b^2 (d+e x)^{5/2} (b d-a e)}{5 e^4}+\frac{2 b (d+e x)^{3/2} (b d-a e)^2}{e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^3}{e^4}+\frac{2 b^3 (d+e x)^{7/2}}{7 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.2202, size = 88, normalized size = 0.92 \[ \frac{2 b^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{4}} + \frac{6 b^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )}{5 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}}{e^{4}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{3}}{e^{4}} \]

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

[Out]

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

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Mathematica [A]  time = 0.0886603, size = 101, normalized size = 1.05 \[ \frac{2 \sqrt{d+e x} \left (35 a^3 e^3+35 a^2 b e^2 (e x-2 d)+7 a b^2 e \left (8 d^2-4 d e x+3 e^2 x^2\right )+b^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )}{35 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(35*a^3*e^3 + 35*a^2*b*e^2*(-2*d + e*x) + 7*a*b^2*e*(8*d^2 - 4*
d*e*x + 3*e^2*x^2) + b^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3)))/(35*e
^4)

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Maple [A]  time = 0.012, size = 116, normalized size = 1.2 \[{\frac{10\,{x}^{3}{b}^{3}{e}^{3}+42\,{x}^{2}a{b}^{2}{e}^{3}-12\,{x}^{2}{b}^{3}d{e}^{2}+70\,x{a}^{2}b{e}^{3}-56\,xa{b}^{2}d{e}^{2}+16\,x{b}^{3}{d}^{2}e+70\,{a}^{3}{e}^{3}-140\,{a}^{2}bd{e}^{2}+112\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{35\,{e}^{4}}\sqrt{ex+d}} \]

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

[Out]

2/35*(5*b^3*e^3*x^3+21*a*b^2*e^3*x^2-6*b^3*d*e^2*x^2+35*a^2*b*e^3*x-28*a*b^2*d*e
^2*x+8*b^3*d^2*e*x+35*a^3*e^3-70*a^2*b*d*e^2+56*a*b^2*d^2*e-16*b^3*d^3)*(e*x+d)^
(1/2)/e^4

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Maxima [A]  time = 0.71676, size = 159, normalized size = 1.66 \[ \frac{2 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{3} - 21 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{e x + d}\right )}}{35 \, e^{4}} \]

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

[Out]

2/35*(5*(e*x + d)^(7/2)*b^3 - 21*(b^3*d - a*b^2*e)*(e*x + d)^(5/2) + 35*(b^3*d^2
- 2*a*b^2*d*e + a^2*b*e^2)*(e*x + d)^(3/2) - 35*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^
2*b*d*e^2 - a^3*e^3)*sqrt(e*x + d))/e^4

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Fricas [A]  time = 0.281297, size = 155, normalized size = 1.61 \[ \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 56 \, a b^{2} d^{2} e - 70 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} +{\left (8 \, b^{3} d^{2} e - 28 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{4}} \]

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

[Out]

2/35*(5*b^3*e^3*x^3 - 16*b^3*d^3 + 56*a*b^2*d^2*e - 70*a^2*b*d*e^2 + 35*a^3*e^3
- 3*(2*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + (8*b^3*d^2*e - 28*a*b^2*d*e^2 + 35*a^2*b*e
^3)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 24.0215, size = 394, normalized size = 4.1 \[ \begin{cases} - \frac{\frac{2 a^{3} d}{\sqrt{d + e x}} + 2 a^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{6 a^{2} b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{6 a^{2} b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{6 a b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{6 a b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 b^{3} d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 b^{3} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{\begin{cases} a^{3} x & \text{for}\: b = 0 \\\frac{a^{3} b x + \frac{3 a^{2} b^{2} x^{2}}{2} + a b^{3} x^{3} + \frac{b^{4} x^{4}}{4}}{b} & \text{otherwise} \end{cases}}{\sqrt{d}} & \text{otherwise} \end{cases} \]

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

[Out]

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

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GIAC/XCAS [A]  time = 0.279817, size = 212, normalized size = 2.21 \[ \frac{2}{35} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{2} b e^{\left (-1\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a b^{2} e^{\left (-10\right )} +{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} b^{3} e^{\left (-21\right )} + 35 \, \sqrt{x e + d} a^{3}\right )} e^{\left (-1\right )} \]

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)/sqrt(e*x + d),x, algorithm="giac")

[Out]

2/35*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*e^(-1) + 7*(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*b^2*e^(-10) + (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)*b^3*e^(-21) + 35*sqrt(x*e + d)*a^3)*e^(-1)