3.2053 \(\int (a+b x) \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=156 \[ -\frac{10 b^4 (d+e x)^{11/2} (b d-a e)}{11 e^6}+\frac{20 b^3 (d+e x)^{9/2} (b d-a e)^2}{9 e^6}-\frac{20 b^2 (d+e x)^{7/2} (b d-a e)^3}{7 e^6}+\frac{2 b (d+e x)^{5/2} (b d-a e)^4}{e^6}-\frac{2 (d+e x)^{3/2} (b d-a e)^5}{3 e^6}+\frac{2 b^5 (d+e x)^{13/2}}{13 e^6} \]

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(3/2))/(3*e^6) + (2*b*(b*d - a*e)^4*(d + e*x)^(5/2))/e^6 - (20*b^2*(b*d - a*e)^3*(
d + e*x)^(7/2))/(7*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(9/2))/(9*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(11/2
))/(11*e^6) + (2*b^5*(d + e*x)^(13/2))/(13*e^6)

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Rubi [A]  time = 0.0538012, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {27, 43} \[ -\frac{10 b^4 (d+e x)^{11/2} (b d-a e)}{11 e^6}+\frac{20 b^3 (d+e x)^{9/2} (b d-a e)^2}{9 e^6}-\frac{20 b^2 (d+e x)^{7/2} (b d-a e)^3}{7 e^6}+\frac{2 b (d+e x)^{5/2} (b d-a e)^4}{e^6}-\frac{2 (d+e x)^{3/2} (b d-a e)^5}{3 e^6}+\frac{2 b^5 (d+e x)^{13/2}}{13 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(3/2))/(3*e^6) + (2*b*(b*d - a*e)^4*(d + e*x)^(5/2))/e^6 - (20*b^2*(b*d - a*e)^3*(
d + e*x)^(7/2))/(7*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(9/2))/(9*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(11/2
))/(11*e^6) + (2*b^5*(d + e*x)^(13/2))/(13*e^6)

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 (a+b x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 \sqrt{d+e x} \, dx\\ &=\int \left (\frac{(-b d+a e)^5 \sqrt{d+e x}}{e^5}+\frac{5 b (b d-a e)^4 (d+e x)^{3/2}}{e^5}-\frac{10 b^2 (b d-a e)^3 (d+e x)^{5/2}}{e^5}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{7/2}}{e^5}-\frac{5 b^4 (b d-a e) (d+e x)^{9/2}}{e^5}+\frac{b^5 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (b d-a e)^5 (d+e x)^{3/2}}{3 e^6}+\frac{2 b (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac{20 b^2 (b d-a e)^3 (d+e x)^{7/2}}{7 e^6}+\frac{20 b^3 (b d-a e)^2 (d+e x)^{9/2}}{9 e^6}-\frac{10 b^4 (b d-a e) (d+e x)^{11/2}}{11 e^6}+\frac{2 b^5 (d+e x)^{13/2}}{13 e^6}\\ \end{align*}

Mathematica [A]  time = 0.121869, size = 123, normalized size = 0.79 \[ \frac{2 (d+e x)^{3/2} \left (-12870 b^2 (d+e x)^2 (b d-a e)^3+10010 b^3 (d+e x)^3 (b d-a e)^2-4095 b^4 (d+e x)^4 (b d-a e)+9009 b (d+e x) (b d-a e)^4-3003 (b d-a e)^5+693 b^5 (d+e x)^5\right )}{9009 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-3003*(b*d - a*e)^5 + 9009*b*(b*d - a*e)^4*(d + e*x) - 12870*b^2*(b*d - a*e)^3*(d + e*x)^2
+ 10010*b^3*(b*d - a*e)^2*(d + e*x)^3 - 4095*b^4*(b*d - a*e)*(d + e*x)^4 + 693*b^5*(d + e*x)^5))/(9009*e^6)

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Maple [B]  time = 0.006, size = 273, normalized size = 1.8 \begin{align*}{\frac{1386\,{x}^{5}{b}^{5}{e}^{5}+8190\,{x}^{4}a{b}^{4}{e}^{5}-1260\,{x}^{4}{b}^{5}d{e}^{4}+20020\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-7280\,{x}^{3}a{b}^{4}d{e}^{4}+1120\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+25740\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-17160\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+6240\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-960\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+18018\,x{a}^{4}b{e}^{5}-20592\,x{a}^{3}{b}^{2}d{e}^{4}+13728\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-4992\,xa{b}^{4}{d}^{3}{e}^{2}+768\,x{b}^{5}{d}^{4}e+6006\,{a}^{5}{e}^{5}-12012\,{a}^{4}bd{e}^{4}+13728\,{a}^{3}{d}^{2}{b}^{2}{e}^{3}-9152\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+3328\,a{d}^{4}{b}^{4}e-512\,{b}^{5}{d}^{5}}{9009\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

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

[Out]

2/9009*(e*x+d)^(3/2)*(693*b^5*e^5*x^5+4095*a*b^4*e^5*x^4-630*b^5*d*e^4*x^4+10010*a^2*b^3*e^5*x^3-3640*a*b^4*d*
e^4*x^3+560*b^5*d^2*e^3*x^3+12870*a^3*b^2*e^5*x^2-8580*a^2*b^3*d*e^4*x^2+3120*a*b^4*d^2*e^3*x^2-480*b^5*d^3*e^
2*x^2+9009*a^4*b*e^5*x-10296*a^3*b^2*d*e^4*x+6864*a^2*b^3*d^2*e^3*x-2496*a*b^4*d^3*e^2*x+384*b^5*d^4*e*x+3003*
a^5*e^5-6006*a^4*b*d*e^4+6864*a^3*b^2*d^2*e^3-4576*a^2*b^3*d^3*e^2+1664*a*b^4*d^4*e-256*b^5*d^5)/e^6

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Maxima [A]  time = 0.979425, size = 350, normalized size = 2.24 \begin{align*} \frac{2 \,{\left (693 \,{\left (e x + d\right )}^{\frac{13}{2}} b^{5} - 4095 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 10010 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 12870 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 3003 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{9009 \, e^{6}} \end{align*}

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

[Out]

2/9009*(693*(e*x + d)^(13/2)*b^5 - 4095*(b^5*d - a*b^4*e)*(e*x + d)^(11/2) + 10010*(b^5*d^2 - 2*a*b^4*d*e + a^
2*b^3*e^2)*(e*x + d)^(9/2) - 12870*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(7/2) +
9009*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(5/2) - 3003*(b^5*
d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x + d)^(3/2))/e^6

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Fricas [B]  time = 1.27672, size = 761, normalized size = 4.88 \begin{align*} \frac{2 \,{\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \,{\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} +{\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt{e x + d}}{9009 \, e^{6}} \end{align*}

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

[Out]

2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*
a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*
a^2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^
5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e
- 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(
e*x + d)/e^6

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Sympy [B]  time = 5.4585, size = 314, normalized size = 2.01 \begin{align*} \frac{2 \left (\frac{b^{5} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (5 a b^{4} e - 5 b^{5} d\right )}{11 e^{5}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (10 a^{2} b^{3} e^{2} - 20 a b^{4} d e + 10 b^{5} d^{2}\right )}{9 e^{5}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (10 a^{3} b^{2} e^{3} - 30 a^{2} b^{3} d e^{2} + 30 a b^{4} d^{2} e - 10 b^{5} d^{3}\right )}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (5 a^{4} b e^{4} - 20 a^{3} b^{2} d e^{3} + 30 a^{2} b^{3} d^{2} e^{2} - 20 a b^{4} d^{3} e + 5 b^{5} d^{4}\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}\right )}{3 e^{5}}\right )}{e} \end{align*}

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

[Out]

2*(b**5*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(5*a*b**4*e - 5*b**5*d)/(11*e**5) + (d + e*x)**(9/2)*(
10*a**2*b**3*e**2 - 20*a*b**4*d*e + 10*b**5*d**2)/(9*e**5) + (d + e*x)**(7/2)*(10*a**3*b**2*e**3 - 30*a**2*b**
3*d*e**2 + 30*a*b**4*d**2*e - 10*b**5*d**3)/(7*e**5) + (d + e*x)**(5/2)*(5*a**4*b*e**4 - 20*a**3*b**2*d*e**3 +
30*a**2*b**3*d**2*e**2 - 20*a*b**4*d**3*e + 5*b**5*d**4)/(5*e**5) + (d + e*x)**(3/2)*(a**5*e**5 - 5*a**4*b*d*
e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d**4*e - b**5*d**5)/(3*e**5))/e

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Giac [B]  time = 1.14503, size = 405, normalized size = 2.6 \begin{align*} \frac{2}{9009} \,{\left (3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{4} b e^{\left (-1\right )} + 858 \,{\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^{3} b^{2} e^{\left (-2\right )} + 286 \,{\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^{2} b^{3} e^{\left (-3\right )} + 13 \,{\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 )} a b^{4} e^{\left (-4\right )} +{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\right )} b^{5} e^{\left (-5\right )} + 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{5}\right )} e^{\left (-1\right )} \end{align*}

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

[Out]

2/9009*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^4*b*e^(-1) + 858*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(
5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3*b^2*e^(-2) + 286*(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^2*b^3*e^(-3) + 13*(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)*a*b^4*e^(-4) + (693*(x*e + d)^
(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2
)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b^5*e^(-5) + 3003*(x*e + d)^(3/2)*a^5)*e^(-1)