3.1632 \(\int \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=129 \[ -\frac{8 b^3 (d+e x)^{9/2} (b d-a e)}{9 e^5}+\frac{12 b^2 (d+e x)^{7/2} (b d-a e)^2}{7 e^5}-\frac{8 b (d+e x)^{5/2} (b d-a e)^3}{5 e^5}+\frac{2 (d+e x)^{3/2} (b d-a e)^4}{3 e^5}+\frac{2 b^4 (d+e x)^{11/2}}{11 e^5} \]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(3/2))/(3*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(5/2))/(5*e^5) + (12*b^2*(b*d - a*e)^
2*(d + e*x)^(7/2))/(7*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^5) + (2*b^4*(d + e*x)^(11/2))/(11*e^5)

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Rubi [A]  time = 0.0540715, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {27, 43} \[ -\frac{8 b^3 (d+e x)^{9/2} (b d-a e)}{9 e^5}+\frac{12 b^2 (d+e x)^{7/2} (b d-a e)^2}{7 e^5}-\frac{8 b (d+e x)^{5/2} (b d-a e)^3}{5 e^5}+\frac{2 (d+e x)^{3/2} (b d-a e)^4}{3 e^5}+\frac{2 b^4 (d+e x)^{11/2}}{11 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(3/2))/(3*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(5/2))/(5*e^5) + (12*b^2*(b*d - a*e)^
2*(d + e*x)^(7/2))/(7*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^5) + (2*b^4*(d + e*x)^(11/2))/(11*e^5)

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 )^2 \, dx &=\int (a+b x)^4 \sqrt{d+e x} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 \sqrt{d+e x}}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^{3/2}}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{5/2}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{7/2}}{e^4}+\frac{b^4 (d+e x)^{9/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^4 (d+e x)^{3/2}}{3 e^5}-\frac{8 b (b d-a e)^3 (d+e x)^{5/2}}{5 e^5}+\frac{12 b^2 (b d-a e)^2 (d+e x)^{7/2}}{7 e^5}-\frac{8 b^3 (b d-a e) (d+e x)^{9/2}}{9 e^5}+\frac{2 b^4 (d+e x)^{11/2}}{11 e^5}\\ \end{align*}

Mathematica [A]  time = 0.0845308, size = 101, normalized size = 0.78 \[ \frac{2 (d+e x)^{3/2} \left (2970 b^2 (d+e x)^2 (b d-a e)^2-1540 b^3 (d+e x)^3 (b d-a e)-2772 b (d+e x) (b d-a e)^3+1155 (b d-a e)^4+315 b^4 (d+e x)^4\right )}{3465 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(1155*(b*d - a*e)^4 - 2772*b*(b*d - a*e)^3*(d + e*x) + 2970*b^2*(b*d - a*e)^2*(d + e*x)^2 -
1540*b^3*(b*d - a*e)*(d + e*x)^3 + 315*b^4*(d + e*x)^4))/(3465*e^5)

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Maple [A]  time = 0.045, size = 186, normalized size = 1.4 \begin{align*}{\frac{630\,{x}^{4}{b}^{4}{e}^{4}+3080\,{x}^{3}a{b}^{3}{e}^{4}-560\,{x}^{3}{b}^{4}d{e}^{3}+5940\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-2640\,{x}^{2}a{b}^{3}d{e}^{3}+480\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+5544\,x{a}^{3}b{e}^{4}-4752\,x{a}^{2}{b}^{2}d{e}^{3}+2112\,xa{b}^{3}{d}^{2}{e}^{2}-384\,x{b}^{4}{d}^{3}e+2310\,{a}^{4}{e}^{4}-3696\,{a}^{3}bd{e}^{3}+3168\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-1408\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{3465\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(e*x+d)^(3/2)*(315*b^4*e^4*x^4+1540*a*b^3*e^4*x^3-280*b^4*d*e^3*x^3+2970*a^2*b^2*e^4*x^2-1320*a*b^3*d*e
^3*x^2+240*b^4*d^2*e^2*x^2+2772*a^3*b*e^4*x-2376*a^2*b^2*d*e^3*x+1056*a*b^3*d^2*e^2*x-192*b^4*d^3*e*x+1155*a^4
*e^4-1848*a^3*b*d*e^3+1584*a^2*b^2*d^2*e^2-704*a*b^3*d^3*e+128*b^4*d^4)/e^5

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Maxima [A]  time = 1.16986, size = 244, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{4} - 1540 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 2970 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 2772 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*(e*x + d)^(11/2)*b^4 - 1540*(b^4*d - a*b^3*e)*(e*x + d)^(9/2) + 2970*(b^4*d^2 - 2*a*b^3*d*e + a^2*
b^2*e^2)*(e*x + d)^(7/2) - 2772*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d)^(5/2) + 1155
*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(e*x + d)^(3/2))/e^5

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Fricas [B]  time = 1.50096, size = 547, normalized size = 4.24 \begin{align*} \frac{2 \,{\left (315 \, b^{4} e^{5} x^{5} + 128 \, b^{4} d^{5} - 704 \, a b^{3} d^{4} e + 1584 \, a^{2} b^{2} d^{3} e^{2} - 1848 \, a^{3} b d^{2} e^{3} + 1155 \, a^{4} d e^{4} + 35 \,{\left (b^{4} d e^{4} + 44 \, a b^{3} e^{5}\right )} x^{4} - 10 \,{\left (4 \, b^{4} d^{2} e^{3} - 22 \, a b^{3} d e^{4} - 297 \, a^{2} b^{2} e^{5}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{3} e^{2} - 44 \, a b^{3} d^{2} e^{3} + 99 \, a^{2} b^{2} d e^{4} + 462 \, a^{3} b e^{5}\right )} x^{2} -{\left (64 \, b^{4} d^{4} e - 352 \, a b^{3} d^{3} e^{2} + 792 \, a^{2} b^{2} d^{2} e^{3} - 924 \, a^{3} b d e^{4} - 1155 \, a^{4} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*b^4*e^5*x^5 + 128*b^4*d^5 - 704*a*b^3*d^4*e + 1584*a^2*b^2*d^3*e^2 - 1848*a^3*b*d^2*e^3 + 1155*a^4
*d*e^4 + 35*(b^4*d*e^4 + 44*a*b^3*e^5)*x^4 - 10*(4*b^4*d^2*e^3 - 22*a*b^3*d*e^4 - 297*a^2*b^2*e^5)*x^3 + 6*(8*
b^4*d^3*e^2 - 44*a*b^3*d^2*e^3 + 99*a^2*b^2*d*e^4 + 462*a^3*b*e^5)*x^2 - (64*b^4*d^4*e - 352*a*b^3*d^3*e^2 + 7
92*a^2*b^2*d^2*e^3 - 924*a^3*b*d*e^4 - 1155*a^4*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 4.66578, size = 223, normalized size = 1.73 \begin{align*} \frac{2 \left (\frac{b^{4} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (4 a b^{3} e - 4 b^{4} d\right )}{9 e^{4}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (4 a^{3} b e^{3} - 12 a^{2} b^{2} d e^{2} + 12 a b^{3} d^{2} e - 4 b^{4} d^{3}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}\right )}{3 e^{4}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b**4*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(9/2)*(4*a*b**3*e - 4*b**4*d)/(9*e**4) + (d + e*x)**(7/2)*(6*
a**2*b**2*e**2 - 12*a*b**3*d*e + 6*b**4*d**2)/(7*e**4) + (d + e*x)**(5/2)*(4*a**3*b*e**3 - 12*a**2*b**2*d*e**2
+ 12*a*b**3*d**2*e - 4*b**4*d**3)/(5*e**4) + (d + e*x)**(3/2)*(a**4*e**4 - 4*a**3*b*d*e**3 + 6*a**2*b**2*d**2
*e**2 - 4*a*b**3*d**3*e + b**4*d**4)/(3*e**4))/e

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Giac [A]  time = 1.19623, size = 292, normalized size = 2.26 \begin{align*} \frac{2}{3465} \,{\left (924 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{3} b e^{\left (-1\right )} + 198 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a^{2} b^{2} e^{\left (-2\right )} + 44 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} a b^{3} e^{\left (-3\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} b^{4} e^{\left (-4\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(924*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*b*e^(-1) + 198*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5
/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b^2*e^(-2) + 44*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d
)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*b^3*e^(-3) + (315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x
*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^4*e^(-4) + 1155*(x*e + d)^(3/2)*a^4
)*e^(-1)