3.2052 \(\int (a+b x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=158 \[ -\frac{10 b^4 (d+e x)^{13/2} (b d-a e)}{13 e^6}+\frac{20 b^3 (d+e x)^{11/2} (b d-a e)^2}{11 e^6}-\frac{20 b^2 (d+e x)^{9/2} (b d-a e)^3}{9 e^6}+\frac{10 b (d+e x)^{7/2} (b d-a e)^4}{7 e^6}-\frac{2 (d+e x)^{5/2} (b d-a e)^5}{5 e^6}+\frac{2 b^5 (d+e x)^{15/2}}{15 e^6} \]

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^6) - (20*b^2*(b*d - a*e
)^3*(d + e*x)^(9/2))/(9*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^6) - (10*b^4*(b*d - a*e)*(d + e*x
)^(13/2))/(13*e^6) + (2*b^5*(d + e*x)^(15/2))/(15*e^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0574865, antiderivative size = 158, 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} \[ -\frac{10 b^4 (d+e x)^{13/2} (b d-a e)}{13 e^6}+\frac{20 b^3 (d+e x)^{11/2} (b d-a e)^2}{11 e^6}-\frac{20 b^2 (d+e x)^{9/2} (b d-a e)^3}{9 e^6}+\frac{10 b (d+e x)^{7/2} (b d-a e)^4}{7 e^6}-\frac{2 (d+e x)^{5/2} (b d-a e)^5}{5 e^6}+\frac{2 b^5 (d+e x)^{15/2}}{15 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^6) - (20*b^2*(b*d - a*e
)^3*(d + e*x)^(9/2))/(9*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^6) - (10*b^4*(b*d - a*e)*(d + e*x
)^(13/2))/(13*e^6) + (2*b^5*(d + e*x)^(15/2))/(15*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) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^5 (d+e x)^{3/2}}{e^5}+\frac{5 b (b d-a e)^4 (d+e x)^{5/2}}{e^5}-\frac{10 b^2 (b d-a e)^3 (d+e x)^{7/2}}{e^5}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{9/2}}{e^5}-\frac{5 b^4 (b d-a e) (d+e x)^{11/2}}{e^5}+\frac{b^5 (d+e x)^{13/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (b d-a e)^5 (d+e x)^{5/2}}{5 e^6}+\frac{10 b (b d-a e)^4 (d+e x)^{7/2}}{7 e^6}-\frac{20 b^2 (b d-a e)^3 (d+e x)^{9/2}}{9 e^6}+\frac{20 b^3 (b d-a e)^2 (d+e x)^{11/2}}{11 e^6}-\frac{10 b^4 (b d-a e) (d+e x)^{13/2}}{13 e^6}+\frac{2 b^5 (d+e x)^{15/2}}{15 e^6}\\ \end{align*}

Mathematica [A]  time = 0.121983, size = 123, normalized size = 0.78 \[ \frac{2 (d+e x)^{5/2} \left (-50050 b^2 (d+e x)^2 (b d-a e)^3+40950 b^3 (d+e x)^3 (b d-a e)^2-17325 b^4 (d+e x)^4 (b d-a e)+32175 b (d+e x) (b d-a e)^4-9009 (b d-a e)^5+3003 b^5 (d+e x)^5\right )}{45045 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(-9009*(b*d - a*e)^5 + 32175*b*(b*d - a*e)^4*(d + e*x) - 50050*b^2*(b*d - a*e)^3*(d + e*x)^
2 + 40950*b^3*(b*d - a*e)^2*(d + e*x)^3 - 17325*b^4*(b*d - a*e)*(d + e*x)^4 + 3003*b^5*(d + e*x)^5))/(45045*e^
6)

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 273, normalized size = 1.7 \begin{align*}{\frac{6006\,{x}^{5}{b}^{5}{e}^{5}+34650\,{x}^{4}a{b}^{4}{e}^{5}-4620\,{x}^{4}{b}^{5}d{e}^{4}+81900\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-25200\,{x}^{3}a{b}^{4}d{e}^{4}+3360\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+100100\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-54600\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+16800\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-2240\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+64350\,x{a}^{4}b{e}^{5}-57200\,x{a}^{3}{b}^{2}d{e}^{4}+31200\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-9600\,xa{b}^{4}{d}^{3}{e}^{2}+1280\,x{b}^{5}{d}^{4}e+18018\,{a}^{5}{e}^{5}-25740\,{a}^{4}bd{e}^{4}+22880\,{a}^{3}{d}^{2}{b}^{2}{e}^{3}-12480\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+3840\,a{d}^{4}{b}^{4}e-512\,{b}^{5}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/45045*(e*x+d)^(5/2)*(3003*b^5*e^5*x^5+17325*a*b^4*e^5*x^4-2310*b^5*d*e^4*x^4+40950*a^2*b^3*e^5*x^3-12600*a*b
^4*d*e^4*x^3+1680*b^5*d^2*e^3*x^3+50050*a^3*b^2*e^5*x^2-27300*a^2*b^3*d*e^4*x^2+8400*a*b^4*d^2*e^3*x^2-1120*b^
5*d^3*e^2*x^2+32175*a^4*b*e^5*x-28600*a^3*b^2*d*e^4*x+15600*a^2*b^3*d^2*e^3*x-4800*a*b^4*d^3*e^2*x+640*b^5*d^4
*e*x+9009*a^5*e^5-12870*a^4*b*d*e^4+11440*a^3*b^2*d^2*e^3-6240*a^2*b^3*d^3*e^2+1920*a*b^4*d^4*e-256*b^5*d^5)/e
^6

________________________________________________________________________________________

Maxima [A]  time = 0.965504, size = 350, normalized size = 2.22 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} b^{5} - 17325 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 40950 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 50050 \,{\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{9}{2}} + 32175 \,{\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{7}{2}} - 9009 \,{\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{5}{2}}\right )}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*b^5 - 17325*(b^5*d - a*b^4*e)*(e*x + d)^(13/2) + 40950*(b^5*d^2 - 2*a*b^4*d*e +
a^2*b^3*e^2)*(e*x + d)^(11/2) - 50050*(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)^(9/
2) + 32175*(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)^(7/2) - 9009*
(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)^(5/2))
/e^6

________________________________________________________________________________________

Fricas [B]  time = 1.25805, size = 953, normalized size = 6.03 \begin{align*} \frac{2 \,{\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \,{\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \,{\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \,{\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} +{\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12
870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e
^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4
+ 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*
(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^
5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2
*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)/e^6

________________________________________________________________________________________

Sympy [A]  time = 22.6336, size = 763, normalized size = 4.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**5*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**5*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 10*a**4*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 10*a**4*b*(d**2*(d + e*x
)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 20*a**3*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d
*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 20*a**3*b**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**3 + 20*a**2*b**3*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**4 + 20*a**2*b**3*(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
+ 10*a*b**4*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**5 + 10*a*b**4*(-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**5*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**6 + 2*b**5*(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**6

________________________________________________________________________________________

Giac [B]  time = 1.15941, size = 919, normalized size = 5.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/45045*(15015*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^4*b*d*e^(-1) + 4290*(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*d*e^(-2) + 1430*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 18
9*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^2*b^3*d*e^(-3) + 65*(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*d*e^(-4) + 5*(
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*d*e^(-5) + 15015*(x*e + d)^(3/2)*a^5*d + 2145*(15*(x*e + d
)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^4*b*e^(-1) + 1430*(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^2*e^(-2) + 130*(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^
3*e^(-3) + 25*(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^4*e^(-4) + (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^5*e^(-5) + 3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)
*a^5)*e^(-1)