3.2057 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=154 \[ -\frac{10 b^4 (d+e x)^{3/2} (b d-a e)}{3 e^6}+\frac{20 b^3 \sqrt{d+e x} (b d-a e)^2}{e^6}+\frac{20 b^2 (b d-a e)^3}{e^6 \sqrt{d+e x}}-\frac{10 b (b d-a e)^4}{3 e^6 (d+e x)^{3/2}}+\frac{2 (b d-a e)^5}{5 e^6 (d+e x)^{5/2}}+\frac{2 b^5 (d+e x)^{5/2}}{5 e^6} \]

[Out]

(2*(b*d - a*e)^5)/(5*e^6*(d + e*x)^(5/2)) - (10*b*(b*d - a*e)^4)/(3*e^6*(d + e*x)^(3/2)) + (20*b^2*(b*d - a*e)
^3)/(e^6*Sqrt[d + e*x]) + (20*b^3*(b*d - a*e)^2*Sqrt[d + e*x])/e^6 - (10*b^4*(b*d - a*e)*(d + e*x)^(3/2))/(3*e
^6) + (2*b^5*(d + e*x)^(5/2))/(5*e^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0548832, antiderivative size = 154, 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} \[ -\frac{10 b^4 (d+e x)^{3/2} (b d-a e)}{3 e^6}+\frac{20 b^3 \sqrt{d+e x} (b d-a e)^2}{e^6}+\frac{20 b^2 (b d-a e)^3}{e^6 \sqrt{d+e x}}-\frac{10 b (b d-a e)^4}{3 e^6 (d+e x)^{3/2}}+\frac{2 (b d-a e)^5}{5 e^6 (d+e x)^{5/2}}+\frac{2 b^5 (d+e x)^{5/2}}{5 e^6} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.103308, size = 123, normalized size = 0.8 \[ \frac{2 \left (150 b^2 (d+e x)^2 (b d-a e)^3+150 b^3 (d+e x)^3 (b d-a e)^2-25 b^4 (d+e x)^4 (b d-a e)-25 b (d+e x) (b d-a e)^4+3 (b d-a e)^5+3 b^5 (d+e x)^5\right )}{15 e^6 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^(7/2),x]

[Out]

(2*(3*(b*d - a*e)^5 - 25*b*(b*d - a*e)^4*(d + e*x) + 150*b^2*(b*d - a*e)^3*(d + e*x)^2 + 150*b^3*(b*d - a*e)^2
*(d + e*x)^3 - 25*b^4*(b*d - a*e)*(d + e*x)^4 + 3*b^5*(d + e*x)^5))/(15*e^6*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.006, size = 273, normalized size = 1.8 \begin{align*} -{\frac{-6\,{x}^{5}{b}^{5}{e}^{5}-50\,{x}^{4}a{b}^{4}{e}^{5}+20\,{x}^{4}{b}^{5}d{e}^{4}-300\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+400\,{x}^{3}a{b}^{4}d{e}^{4}-160\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+300\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-1800\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+2400\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-960\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+50\,x{a}^{4}b{e}^{5}+400\,x{a}^{3}{b}^{2}d{e}^{4}-2400\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+3200\,xa{b}^{4}{d}^{3}{e}^{2}-1280\,x{b}^{5}{d}^{4}e+6\,{a}^{5}{e}^{5}+20\,{a}^{4}bd{e}^{4}+160\,{a}^{3}{d}^{2}{b}^{2}{e}^{3}-960\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+1280\,a{d}^{4}{b}^{4}e-512\,{b}^{5}{d}^{5}}{15\,{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)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(7/2),x)

[Out]

-2/15*(-3*b^5*e^5*x^5-25*a*b^4*e^5*x^4+10*b^5*d*e^4*x^4-150*a^2*b^3*e^5*x^3+200*a*b^4*d*e^4*x^3-80*b^5*d^2*e^3
*x^3+150*a^3*b^2*e^5*x^2-900*a^2*b^3*d*e^4*x^2+1200*a*b^4*d^2*e^3*x^2-480*b^5*d^3*e^2*x^2+25*a^4*b*e^5*x+200*a
^3*b^2*d*e^4*x-1200*a^2*b^3*d^2*e^3*x+1600*a*b^4*d^3*e^2*x-640*b^5*d^4*e*x+3*a^5*e^5+10*a^4*b*d*e^4+80*a^3*b^2
*d^2*e^3-480*a^2*b^3*d^3*e^2+640*a*b^4*d^4*e-256*b^5*d^5)/(e*x+d)^(5/2)/e^6

________________________________________________________________________________________

Maxima [A]  time = 1.03527, size = 358, normalized size = 2.32 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} b^{5} - 25 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 150 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \sqrt{e x + d}}{e^{5}} + \frac{3 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 30 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} + 150 \,{\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 )}^{2} - 25 \,{\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 )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{5}}\right )}}{15 \, 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)^(7/2),x, algorithm="maxima")

[Out]

2/15*((3*(e*x + d)^(5/2)*b^5 - 25*(b^5*d - a*b^4*e)*(e*x + d)^(3/2) + 150*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2
)*sqrt(e*x + d))/e^5 + (3*b^5*d^5 - 15*a*b^4*d^4*e + 30*a^2*b^3*d^3*e^2 - 30*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4
- 3*a^5*e^5 + 150*(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)^2 - 25*(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))/((e*x + d)^(5/2)*e^5))/e

________________________________________________________________________________________

Fricas [B]  time = 1.32887, size = 621, normalized size = 4.03 \begin{align*} \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\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)^(7/2),x, algorithm="fricas")

[Out]

2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 +
30*(16*b^5*d^3*e^2 - 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x^2 + 5*(128*b^5*d^4*e - 320*a*b^4*
d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^
2*e^7*x + d^3*e^6)

________________________________________________________________________________________

Sympy [A]  time = 4.29515, size = 1428, normalized size = 9.27 \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)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*a**5*e**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x))
- 20*a**4*b*d*e**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 50*
a**4*b*e**5*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 160*a**3
*b**2*d**2*e**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 400*a*
*3*b**2*d*e**4*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 300*a
**3*b**2*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 960
*a**2*b**3*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 2
400*a**2*b**3*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)
) + 1800*a**2*b**3*d*e**4*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d +
e*x)) + 300*a**2*b**3*e**5*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d
+ e*x)) - 1280*a*b**4*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d +
e*x)) - 3200*a*b**4*d**3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d
+ e*x)) - 2400*a*b**4*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sq
rt(d + e*x)) - 400*a*b**4*d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*s
qrt(d + e*x)) + 50*a*b**4*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqr
t(d + e*x)) + 512*b**5*d**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*
x)) + 1280*b**5*d**4*e*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x))
+ 960*b**5*d**3*e**2*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x
)) + 160*b**5*d**2*e**3*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e
*x)) - 20*b**5*d*e**4*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x
)) + 6*b**5*e**5*x**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)), N
e(e, 0)), ((a**5*x + 5*a**4*b*x**2/2 + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**4/2 + a*b**4*x**5 + b**5*x**6/6)/d
**(7/2), True))

________________________________________________________________________________________

Giac [B]  time = 1.15558, size = 450, normalized size = 2.92 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{24} - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{24} + 150 \, \sqrt{x e + d} b^{5} d^{2} e^{24} + 25 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} e^{25} - 300 \, \sqrt{x e + d} a b^{4} d e^{25} + 150 \, \sqrt{x e + d} a^{2} b^{3} e^{26}\right )} e^{\left (-30\right )} + \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} b^{5} d^{3} - 25 \,{\left (x e + d\right )} b^{5} d^{4} + 3 \, b^{5} d^{5} - 450 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e + 100 \,{\left (x e + d\right )} a b^{4} d^{3} e - 15 \, a b^{4} d^{4} e + 450 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} - 150 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} + 30 \, a^{2} b^{3} d^{3} e^{2} - 150 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} + 100 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3} - 30 \, a^{3} b^{2} d^{2} e^{3} - 25 \,{\left (x e + d\right )} a^{4} b e^{4} + 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5}\right )} e^{\left (-6\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \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)^(7/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*b^5*e^24 - 25*(x*e + d)^(3/2)*b^5*d*e^24 + 150*sqrt(x*e + d)*b^5*d^2*e^24 + 25*(x*e +
d)^(3/2)*a*b^4*e^25 - 300*sqrt(x*e + d)*a*b^4*d*e^25 + 150*sqrt(x*e + d)*a^2*b^3*e^26)*e^(-30) + 2/15*(150*(x*
e + d)^2*b^5*d^3 - 25*(x*e + d)*b^5*d^4 + 3*b^5*d^5 - 450*(x*e + d)^2*a*b^4*d^2*e + 100*(x*e + d)*a*b^4*d^3*e
- 15*a*b^4*d^4*e + 450*(x*e + d)^2*a^2*b^3*d*e^2 - 150*(x*e + d)*a^2*b^3*d^2*e^2 + 30*a^2*b^3*d^3*e^2 - 150*(x
*e + d)^2*a^3*b^2*e^3 + 100*(x*e + d)*a^3*b^2*d*e^3 - 30*a^3*b^2*d^2*e^3 - 25*(x*e + d)*a^4*b*e^4 + 15*a^4*b*d
*e^4 - 3*a^5*e^5)*e^(-6)/(x*e + d)^(5/2)