3.2056 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{2 b^4 (d+e x)^{5/2} (b d-a e)}{e^6}+\frac{20 b^3 (d+e x)^{3/2} (b d-a e)^2}{3 e^6}-\frac{20 b^2 \sqrt{d+e x} (b d-a e)^3}{e^6}-\frac{10 b (b d-a e)^4}{e^6 \sqrt{d+e x}}+\frac{2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}+\frac{2 b^5 (d+e x)^{7/2}}{7 e^6} \]

[Out]

(2*(b*d - a*e)^5)/(3*e^6*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^4)/(e^6*Sqrt[d + e*x]) - (20*b^2*(b*d - a*e)^3*S
qrt[d + e*x])/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x)^(3/2))/(3*e^6) - (2*b^4*(b*d - a*e)*(d + e*x)^(5/2))/e^6 +
(2*b^5*(d + e*x)^(7/2))/(7*e^6)

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Rubi [A]  time = 0.0553517, antiderivative size = 152, 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} \[ -\frac{2 b^4 (d+e x)^{5/2} (b d-a e)}{e^6}+\frac{20 b^3 (d+e x)^{3/2} (b d-a e)^2}{3 e^6}-\frac{20 b^2 \sqrt{d+e x} (b d-a e)^3}{e^6}-\frac{10 b (b d-a e)^4}{e^6 \sqrt{d+e x}}+\frac{2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}+\frac{2 b^5 (d+e x)^{7/2}}{7 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.101687, size = 123, normalized size = 0.81 \[ \frac{2 \left (-210 b^2 (d+e x)^2 (b d-a e)^3+70 b^3 (d+e x)^3 (b d-a e)^2-21 b^4 (d+e x)^4 (b d-a e)-105 b (d+e x) (b d-a e)^4+7 (b d-a e)^5+3 b^5 (d+e x)^5\right )}{21 e^6 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(7*(b*d - a*e)^5 - 105*b*(b*d - a*e)^4*(d + e*x) - 210*b^2*(b*d - a*e)^3*(d + e*x)^2 + 70*b^3*(b*d - a*e)^2
*(d + e*x)^3 - 21*b^4*(b*d - a*e)*(d + e*x)^4 + 3*b^5*(d + e*x)^5))/(21*e^6*(d + e*x)^(3/2))

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Maple [B]  time = 0.007, size = 273, normalized size = 1.8 \begin{align*} -{\frac{-6\,{x}^{5}{b}^{5}{e}^{5}-42\,{x}^{4}a{b}^{4}{e}^{5}+12\,{x}^{4}{b}^{5}d{e}^{4}-140\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+112\,{x}^{3}a{b}^{4}d{e}^{4}-32\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-420\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+840\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-672\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+192\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+210\,x{a}^{4}b{e}^{5}-1680\,x{a}^{3}{b}^{2}d{e}^{4}+3360\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-2688\,xa{b}^{4}{d}^{3}{e}^{2}+768\,x{b}^{5}{d}^{4}e+14\,{a}^{5}{e}^{5}+140\,{a}^{4}bd{e}^{4}-1120\,{a}^{3}{d}^{2}{b}^{2}{e}^{3}+2240\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-1792\,a{d}^{4}{b}^{4}e+512\,{b}^{5}{d}^{5}}{21\,{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)^(5/2),x)

[Out]

-2/21*(-3*b^5*e^5*x^5-21*a*b^4*e^5*x^4+6*b^5*d*e^4*x^4-70*a^2*b^3*e^5*x^3+56*a*b^4*d*e^4*x^3-16*b^5*d^2*e^3*x^
3-210*a^3*b^2*e^5*x^2+420*a^2*b^3*d*e^4*x^2-336*a*b^4*d^2*e^3*x^2+96*b^5*d^3*e^2*x^2+105*a^4*b*e^5*x-840*a^3*b
^2*d*e^4*x+1680*a^2*b^3*d^2*e^3*x-1344*a*b^4*d^3*e^2*x+384*b^5*d^4*e*x+7*a^5*e^5+70*a^4*b*d*e^4-560*a^3*b^2*d^
2*e^3+1120*a^2*b^3*d^3*e^2-896*a*b^4*d^4*e+256*b^5*d^5)/(e*x+d)^(3/2)/e^6

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Maxima [A]  time = 0.999581, size = 358, normalized size = 2.36 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{5} - 21 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 70 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 210 \,{\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 )} \sqrt{e x + d}}{e^{5}} + \frac{7 \,{\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} - 15 \,{\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 )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{5}}\right )}}{21 \, 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)^(5/2),x, algorithm="maxima")

[Out]

2/21*((3*(e*x + d)^(7/2)*b^5 - 21*(b^5*d - a*b^4*e)*(e*x + d)^(5/2) + 70*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)
*(e*x + d)^(3/2) - 210*(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)*sqrt(e*x + d))/e^5 + 7*(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 - 15*(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)^(3/2)*e^5))/e

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Fricas [B]  time = 1.27831, size = 603, normalized size = 3.97 \begin{align*} \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \,{\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{21 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*
e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3
- 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e - 448*a*b^4
*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d^2
*e^6)

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Sympy [A]  time = 42.7339, size = 196, normalized size = 1.29 \begin{align*} \frac{2 b^{5} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{6}} - \frac{10 b \left (a e - b d\right )^{4}}{e^{6} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (10 a b^{4} e - 10 b^{5} d\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (20 a^{2} b^{3} e^{2} - 40 a b^{4} d e + 20 b^{5} d^{2}\right )}{3 e^{6}} + \frac{\sqrt{d + e x} \left (20 a^{3} b^{2} e^{3} - 60 a^{2} b^{3} d e^{2} + 60 a b^{4} d^{2} e - 20 b^{5} d^{3}\right )}{e^{6}} - \frac{2 \left (a e - b d\right )^{5}}{3 e^{6} \left (d + e x\right )^{\frac{3}{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)**(5/2),x)

[Out]

2*b**5*(d + e*x)**(7/2)/(7*e**6) - 10*b*(a*e - b*d)**4/(e**6*sqrt(d + e*x)) + (d + e*x)**(5/2)*(10*a*b**4*e -
10*b**5*d)/(5*e**6) + (d + e*x)**(3/2)*(20*a**2*b**3*e**2 - 40*a*b**4*d*e + 20*b**5*d**2)/(3*e**6) + sqrt(d +
e*x)*(20*a**3*b**2*e**3 - 60*a**2*b**3*d*e**2 + 60*a*b**4*d**2*e - 20*b**5*d**3)/e**6 - 2*(a*e - b*d)**5/(3*e*
*6*(d + e*x)**(3/2))

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Giac [B]  time = 1.16303, size = 451, normalized size = 2.97 \begin{align*} \frac{2}{21} \,{\left (3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} e^{36} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d e^{36} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{2} e^{36} - 210 \, \sqrt{x e + d} b^{5} d^{3} e^{36} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} e^{37} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d e^{37} + 630 \, \sqrt{x e + d} a b^{4} d^{2} e^{37} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} e^{38} - 630 \, \sqrt{x e + d} a^{2} b^{3} d e^{38} + 210 \, \sqrt{x e + d} a^{3} b^{2} e^{39}\right )} e^{\left (-42\right )} - \frac{2 \,{\left (15 \,{\left (x e + d\right )} b^{5} d^{4} - b^{5} d^{5} - 60 \,{\left (x e + d\right )} a b^{4} d^{3} e + 5 \, a b^{4} d^{4} e + 90 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} - 10 \, a^{2} b^{3} d^{3} e^{2} - 60 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3} + 10 \, a^{3} b^{2} d^{2} e^{3} + 15 \,{\left (x e + d\right )} a^{4} b e^{4} - 5 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

2/21*(3*(x*e + d)^(7/2)*b^5*e^36 - 21*(x*e + d)^(5/2)*b^5*d*e^36 + 70*(x*e + d)^(3/2)*b^5*d^2*e^36 - 210*sqrt(
x*e + d)*b^5*d^3*e^36 + 21*(x*e + d)^(5/2)*a*b^4*e^37 - 140*(x*e + d)^(3/2)*a*b^4*d*e^37 + 630*sqrt(x*e + d)*a
*b^4*d^2*e^37 + 70*(x*e + d)^(3/2)*a^2*b^3*e^38 - 630*sqrt(x*e + d)*a^2*b^3*d*e^38 + 210*sqrt(x*e + d)*a^3*b^2
*e^39)*e^(-42) - 2/3*(15*(x*e + d)*b^5*d^4 - b^5*d^5 - 60*(x*e + d)*a*b^4*d^3*e + 5*a*b^4*d^4*e + 90*(x*e + d)
*a^2*b^3*d^2*e^2 - 10*a^2*b^3*d^3*e^2 - 60*(x*e + d)*a^3*b^2*d*e^3 + 10*a^3*b^2*d^2*e^3 + 15*(x*e + d)*a^4*b*e
^4 - 5*a^4*b*d*e^4 + a^5*e^5)*e^(-6)/(x*e + d)^(3/2)