3.1630 \(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=129 \[ -\frac{8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac{12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac{8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac{2 b^4 (d+e x)^{15/2}}{15 e^5} \]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^5) + (12*b^2*(b*d - a*e)^
2*(d + e*x)^(11/2))/(11*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^4*(d + e*x)^(15/2))/(15*e^
5)

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Rubi [A]  time = 0.041644, 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} \[ -\frac{8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac{12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac{8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac{2 b^4 (d+e x)^{15/2}}{15 e^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^5) + (12*b^2*(b*d - a*e)^
2*(d + e*x)^(11/2))/(11*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^4*(d + e*x)^(15/2))/(15*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 (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (d+e x)^{5/2}}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^{7/2}}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{9/2}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{11/2}}{e^4}+\frac{b^4 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^4 (d+e x)^{7/2}}{7 e^5}-\frac{8 b (b d-a e)^3 (d+e x)^{9/2}}{9 e^5}+\frac{12 b^2 (b d-a e)^2 (d+e x)^{11/2}}{11 e^5}-\frac{8 b^3 (b d-a e) (d+e x)^{13/2}}{13 e^5}+\frac{2 b^4 (d+e x)^{15/2}}{15 e^5}\\ \end{align*}

Mathematica [A]  time = 0.094288, size = 101, normalized size = 0.78 \[ \frac{2 (d+e x)^{7/2} \left (24570 b^2 (d+e x)^2 (b d-a e)^2-13860 b^3 (d+e x)^3 (b d-a e)-20020 b (d+e x) (b d-a e)^3+6435 (b d-a e)^4+3003 b^4 (d+e x)^4\right )}{45045 e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(7/2)*(6435*(b*d - a*e)^4 - 20020*b*(b*d - a*e)^3*(d + e*x) + 24570*b^2*(b*d - a*e)^2*(d + e*x)^2
- 13860*b^3*(b*d - a*e)*(d + e*x)^3 + 3003*b^4*(d + e*x)^4))/(45045*e^5)

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Maple [A]  time = 0.046, size = 186, normalized size = 1.4 \begin{align*}{\frac{6006\,{x}^{4}{b}^{4}{e}^{4}+27720\,{x}^{3}a{b}^{3}{e}^{4}-3696\,{x}^{3}{b}^{4}d{e}^{3}+49140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-15120\,{x}^{2}a{b}^{3}d{e}^{3}+2016\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+40040\,x{a}^{3}b{e}^{4}-21840\,x{a}^{2}{b}^{2}d{e}^{3}+6720\,xa{b}^{3}{d}^{2}{e}^{2}-896\,x{b}^{4}{d}^{3}e+12870\,{a}^{4}{e}^{4}-11440\,{a}^{3}bd{e}^{3}+6240\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-1920\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/45045*(e*x+d)^(7/2)*(3003*b^4*e^4*x^4+13860*a*b^3*e^4*x^3-1848*b^4*d*e^3*x^3+24570*a^2*b^2*e^4*x^2-7560*a*b^
3*d*e^3*x^2+1008*b^4*d^2*e^2*x^2+20020*a^3*b*e^4*x-10920*a^2*b^2*d*e^3*x+3360*a*b^3*d^2*e^2*x-448*b^4*d^3*e*x+
6435*a^4*e^4-5720*a^3*b*d*e^3+3120*a^2*b^2*d^2*e^2-960*a*b^3*d^3*e+128*b^4*d^4)/e^5

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Maxima [A]  time = 1.04471, size = 244, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} b^{4} - 13860 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 24570 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 20020 \,{\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{9}{2}} + 6435 \,{\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{7}{2}}\right )}}{45045 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*b^4 - 13860*(b^4*d - a*b^3*e)*(e*x + d)^(13/2) + 24570*(b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*(e*x + d)^(11/2) - 20020*(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)^(9/2)
+ 6435*(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)^(7/2))/e^5

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Fricas [B]  time = 1.54051, size = 855, normalized size = 6.63 \begin{align*} \frac{2 \,{\left (3003 \, b^{4} e^{7} x^{7} + 128 \, b^{4} d^{7} - 960 \, a b^{3} d^{6} e + 3120 \, a^{2} b^{2} d^{5} e^{2} - 5720 \, a^{3} b d^{4} e^{3} + 6435 \, a^{4} d^{3} e^{4} + 231 \,{\left (31 \, b^{4} d e^{6} + 60 \, a b^{3} e^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{4} d^{2} e^{5} + 540 \, a b^{3} d e^{6} + 390 \, a^{2} b^{2} e^{7}\right )} x^{5} + 35 \,{\left (b^{4} d^{3} e^{4} + 636 \, a b^{3} d^{2} e^{5} + 1794 \, a^{2} b^{2} d e^{6} + 572 \, a^{3} b e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{4} d^{4} e^{3} - 60 \, a b^{3} d^{3} e^{4} - 8814 \, a^{2} b^{2} d^{2} e^{5} - 10868 \, a^{3} b d e^{6} - 1287 \, a^{4} e^{7}\right )} x^{3} + 3 \,{\left (16 \, b^{4} d^{5} e^{2} - 120 \, a b^{3} d^{4} e^{3} + 390 \, a^{2} b^{2} d^{3} e^{4} + 14300 \, a^{3} b d^{2} e^{5} + 6435 \, a^{4} d e^{6}\right )} x^{2} -{\left (64 \, b^{4} d^{6} e - 480 \, a b^{3} d^{5} e^{2} + 1560 \, a^{2} b^{2} d^{4} e^{3} - 2860 \, a^{3} b d^{3} e^{4} - 19305 \, a^{4} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

2/45045*(3003*b^4*e^7*x^7 + 128*b^4*d^7 - 960*a*b^3*d^6*e + 3120*a^2*b^2*d^5*e^2 - 5720*a^3*b*d^4*e^3 + 6435*a
^4*d^3*e^4 + 231*(31*b^4*d*e^6 + 60*a*b^3*e^7)*x^6 + 63*(71*b^4*d^2*e^5 + 540*a*b^3*d*e^6 + 390*a^2*b^2*e^7)*x
^5 + 35*(b^4*d^3*e^4 + 636*a*b^3*d^2*e^5 + 1794*a^2*b^2*d*e^6 + 572*a^3*b*e^7)*x^4 - 5*(8*b^4*d^4*e^3 - 60*a*b
^3*d^3*e^4 - 8814*a^2*b^2*d^2*e^5 - 10868*a^3*b*d*e^6 - 1287*a^4*e^7)*x^3 + 3*(16*b^4*d^5*e^2 - 120*a*b^3*d^4*
e^3 + 390*a^2*b^2*d^3*e^4 + 14300*a^3*b*d^2*e^5 + 6435*a^4*d*e^6)*x^2 - (64*b^4*d^6*e - 480*a*b^3*d^5*e^2 + 15
60*a^2*b^2*d^4*e^3 - 2860*a^3*b*d^3*e^4 - 19305*a^4*d^2*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 33.5058, size = 960, normalized size = 7.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**4*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**4*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**4*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
8*a**3*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 16*a**3*b*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(
d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 8*a**3*b*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5
- 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 12*a**2*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*
x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 24*a**2*b**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5
- 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 12*a**2*b**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)
**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 8*a*b**3*d**2*(-
d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 16*a
*b**3*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2
)/9 + (d + e*x)**(11/2)/11)/e**4 + 8*a*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e
*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 2*b**4*d**
2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 +
(d + e*x)**(11/2)/11)/e**5 + 4*b**4*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(
7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 2*b**4*(d**6*(d
+ e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**
2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5

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Giac [B]  time = 1.25363, size = 1157, normalized size = 8.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/45045*(12012*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*b*d^2*e^(-1) + 2574*(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*d^2*e^(-2) + 572*(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*d^2*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)*b^4*d^2*e^(-4) +
15015*(x*e + d)^(3/2)*a^4*d^2 + 3432*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3*
b*d*e^(-1) + 1716*(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^2*d*e^(-2) + 104*(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^3*d*e^(-3) + 10*(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^4*d*e^(-4) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^4*d + 572*(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^3*b*e^(-1) + 78*(315*(x*e + d)^(11/2) - 1
540*(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^2*b^
2*e^(-2) + 20*(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)*a*b^3*e^(-3) + (3003*(x*e + d)^(15/2) - 20790*(
x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 5405
4*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*b^4*e^(-4) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d
+ 35*(x*e + d)^(3/2)*d^2)*a^4)*e^(-1)