3.1624 \(\int \sqrt {d+e x} (a^2+2 a b x+b^2 x^2) \, dx\)

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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \[ -\frac {4 b (d+e x)^{5/2} (b d-a e)}{5 e^3}+\frac {2 (d+e x)^{3/2} (b d-a e)^2}{3 e^3}+\frac {2 b^2 (d+e x)^{7/2}}{7 e^3} \]

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)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^2 \sqrt {d+e x}}{e^2}-\frac {2 b (b d-a e) (d+e x)^{3/2}}{e^2}+\frac {b^2 (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e)^2 (d+e x)^{3/2}}{3 e^3}-\frac {4 b (b d-a e) (d+e x)^{5/2}}{5 e^3}+\frac {2 b^2 (d+e x)^{7/2}}{7 e^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, 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)

________________________________________________________________________________________

fricas [A]  time = 1.05, size = 99, normalized size = 1.39 \[ \frac {2 \, {\left (15 \, b^{2} e^{3} x^{3} + 8 \, b^{2} d^{3} - 28 \, a b d^{2} e + 35 \, a^{2} d e^{2} + 3 \, {\left (b^{2} d e^{2} + 14 \, a b e^{3}\right )} x^{2} - {\left (4 \, b^{2} d^{2} e - 14 \, a b d e^{2} - 35 \, a^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

giac [B]  time = 0.18, size = 210, normalized size = 2.96 \[ \frac {2}{105} \, {\left (70 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b d e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b^{2} d e^{\left (-2\right )} + 14 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a b e^{\left (-1\right )} + 3 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b^{2} e^{\left (-2\right )} + 105 \, \sqrt {x e + d} a^{2} d + 35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/105*(70*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*d*e^(-1) + 7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 1
5*sqrt(x*e + d)*d^2)*b^2*d*e^(-2) + 14*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b*e
^(-1) + 3*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*b^2*e^(-2
) + 105*sqrt(x*e + d)*a^2*d + 35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2)*e^(-1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 63, normalized size = 0.89 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (15 b^{2} e^{2} x^{2}+42 a b \,e^{2} x -12 b^{2} d e x +35 a^{2} e^{2}-28 a b d e +8 b^{2} d^{2}\right )}{105 e^{3}} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.02, size = 68, normalized size = 0.96 \[ \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{2} - 42 \, {\left (b^{2} d - a b e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{105 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

mupad [B]  time = 0.56, size = 68, normalized size = 0.96 \[ \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (15\,b^2\,{\left (d+e\,x\right )}^2+35\,a^2\,e^2+35\,b^2\,d^2-42\,b^2\,d\,\left (d+e\,x\right )+42\,a\,b\,e\,\left (d+e\,x\right )-70\,a\,b\,d\,e\right )}{105\,e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 3.49, size = 85, normalized size = 1.20 \[ \frac {2 \left (\frac {b^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 a b e - 2 b^{2} d\right )}{5 e^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} e^{2} - 2 a b d e + b^{2} d^{2}\right )}{3 e^{2}}\right )}{e} \]

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

________________________________________________________________________________________