3.1633 \(\int \frac{(a^2+2 a b x+b^2 x^2)^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=127 \[ -\frac{8 b^3 (d+e x)^{7/2} (b d-a e)}{7 e^5}+\frac{12 b^2 (d+e x)^{5/2} (b d-a e)^2}{5 e^5}-\frac{8 b (d+e x)^{3/2} (b d-a e)^3}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^4}{e^5}+\frac{2 b^4 (d+e x)^{9/2}}{9 e^5} \]

[Out]

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

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Rubi [A]  time = 0.0414936, antiderivative size = 127, 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} \[ -\frac{8 b^3 (d+e x)^{7/2} (b d-a e)}{7 e^5}+\frac{12 b^2 (d+e x)^{5/2} (b d-a e)^2}{5 e^5}-\frac{8 b (d+e x)^{3/2} (b d-a e)^3}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^4}{e^5}+\frac{2 b^4 (d+e x)^{9/2}}{9 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0706069, size = 101, normalized size = 0.8 \[ \frac{2 \sqrt{d+e x} \left (378 b^2 (d+e x)^2 (b d-a e)^2-180 b^3 (d+e x)^3 (b d-a e)-420 b (d+e x) (b d-a e)^3+315 (b d-a e)^4+35 b^4 (d+e x)^4\right )}{315 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(315*(b*d - a*e)^4 - 420*b*(b*d - a*e)^3*(d + e*x) + 378*b^2*(b*d - a*e)^2*(d + e*x)^2 - 180*
b^3*(b*d - a*e)*(d + e*x)^3 + 35*b^4*(d + e*x)^4))/(315*e^5)

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Maple [A]  time = 0.044, size = 186, normalized size = 1.5 \begin{align*}{\frac{70\,{x}^{4}{b}^{4}{e}^{4}+360\,{x}^{3}a{b}^{3}{e}^{4}-80\,{x}^{3}{b}^{4}d{e}^{3}+756\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-432\,{x}^{2}a{b}^{3}d{e}^{3}+96\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+840\,x{a}^{3}b{e}^{4}-1008\,x{a}^{2}{b}^{2}d{e}^{3}+576\,xa{b}^{3}{d}^{2}{e}^{2}-128\,x{b}^{4}{d}^{3}e+630\,{a}^{4}{e}^{4}-1680\,{a}^{3}bd{e}^{3}+2016\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-1152\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \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]

2/315*(35*b^4*e^4*x^4+180*a*b^3*e^4*x^3-40*b^4*d*e^3*x^3+378*a^2*b^2*e^4*x^2-216*a*b^3*d*e^3*x^2+48*b^4*d^2*e^
2*x^2+420*a^3*b*e^4*x-504*a^2*b^2*d*e^3*x+288*a*b^3*d^2*e^2*x-64*b^4*d^3*e*x+315*a^4*e^4-840*a^3*b*d*e^3+1008*
a^2*b^2*d^2*e^2-576*a*b^3*d^3*e+128*b^4*d^4)*(e*x+d)^(1/2)/e^5

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Maxima [B]  time = 1.17734, size = 333, normalized size = 2.62 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{4} + 42 \,{\left (\frac{10 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )} a b}{e} + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )} a^{2} + \frac{84 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} a^{2} b^{2}}{e^{2}} + \frac{36 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )} a b^{3}}{e^{3}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} b^{4}}{e^{4}}\right )}}{315 \, 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, algorithm="maxima")

[Out]

2/315*(315*sqrt(e*x + d)*a^4 + 42*(10*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b/e + (3*(e*x + d)^(5/2) - 10*(e
*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2/e^2)*a^2 + 84*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(
e*x + d)*d^2)*a^2*b^2/e^2 + 36*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*
x + d)*d^3)*a*b^3/e^3 + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^
(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^4/e^4)/e

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Fricas [A]  time = 1.49323, size = 405, normalized size = 3.19 \begin{align*} \frac{2 \,{\left (35 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 576 \, a b^{3} d^{3} e + 1008 \, a^{2} b^{2} d^{2} e^{2} - 840 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - 20 \,{\left (2 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{2} e^{2} - 36 \, a b^{3} d e^{3} + 63 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{3} e - 72 \, a b^{3} d^{2} e^{2} + 126 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, 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]

2/315*(35*b^4*e^4*x^4 + 128*b^4*d^4 - 576*a*b^3*d^3*e + 1008*a^2*b^2*d^2*e^2 - 840*a^3*b*d*e^3 + 315*a^4*e^4 -
20*(2*b^4*d*e^3 - 9*a*b^3*e^4)*x^3 + 6*(8*b^4*d^2*e^2 - 36*a*b^3*d*e^3 + 63*a^2*b^2*e^4)*x^2 - 4*(16*b^4*d^3*
e - 72*a*b^3*d^2*e^2 + 126*a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 61.6239, size = 561, normalized size = 4.42 \begin{align*} \begin{cases} - \frac{\frac{2 a^{4} d}{\sqrt{d + e x}} + 2 a^{4} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{8 a^{3} b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{8 a^{3} 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{12 a^{2} 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{12 a^{2} 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{8 a 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{8 a 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}} + \frac{2 b^{4} d \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^{4}} + \frac{2 b^{4} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac{b^{4} x^{5}}{5}}{\sqrt{d}} & \text{otherwise} \end{cases} \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]

Piecewise((-(2*a**4*d/sqrt(d + e*x) + 2*a**4*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 8*a**3*b*d*(-d/sqrt(d + e*x)
- sqrt(d + e*x))/e + 8*a**3*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 12*a**2*b**2*
d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 12*a**2*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 + 8*a*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*s
qrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 + 8*a*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 + 2*b**4*d*(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**4 + 2*b
**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(
d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/e, Ne(e, 0)), ((a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2*x**3 + a*b*
*3*x**4 + b**4*x**5/5)/sqrt(d), True))

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Giac [A]  time = 1.20947, size = 289, normalized size = 2.28 \begin{align*} \frac{2}{315} \,{\left (420 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{3} b e^{\left (-1\right )} + 126 \,{\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^{2} b^{2} e^{\left (-2\right )} + 36 \,{\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 )} a b^{3} e^{\left (-3\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} b^{4} e^{\left (-4\right )} + 315 \, \sqrt{x e + d} 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]

2/315*(420*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*e^(-1) + 126*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d
+ 15*sqrt(x*e + d)*d^2)*a^2*b^2*e^(-2) + 36*(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)*a*b^3*e^(-3) + (35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2
- 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^4*e^(-4) + 315*sqrt(x*e + d)*a^4)*e^(-1)