3.2048 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{6 b^2 \sqrt{d+e x} (b d-a e)}{e^4}-\frac{6 b (b d-a e)^2}{e^4 \sqrt{d+e x}}+\frac{2 (b d-a e)^3}{3 e^4 (d+e x)^{3/2}}+\frac{2 b^3 (d+e x)^{3/2}}{3 e^4} \]

[Out]

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

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Rubi [A]  time = 0.0332519, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{6 b^2 \sqrt{d+e x} (b d-a e)}{e^4}-\frac{6 b (b d-a e)^2}{e^4 \sqrt{d+e x}}+\frac{2 (b d-a e)^3}{3 e^4 (d+e x)^{3/2}}+\frac{2 b^3 (d+e x)^{3/2}}{3 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0531041, size = 76, normalized size = 0.79 \[ \frac{2 \left (-9 b^2 (d+e x)^2 (b d-a e)-9 b (d+e x) (b d-a e)^2+(b d-a e)^3+b^3 (d+e x)^3\right )}{3 e^4 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((b*d - a*e)^3 - 9*b*(b*d - a*e)^2*(d + e*x) - 9*b^2*(b*d - a*e)*(d + e*x)^2 + b^3*(d + e*x)^3))/(3*e^4*(d
+ e*x)^(3/2))

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Maple [A]  time = 0.008, size = 115, normalized size = 1.2 \begin{align*} -{\frac{-2\,{x}^{3}{b}^{3}{e}^{3}-18\,{x}^{2}a{b}^{2}{e}^{3}+12\,{x}^{2}{b}^{3}d{e}^{2}+18\,x{a}^{2}b{e}^{3}-72\,xa{b}^{2}d{e}^{2}+48\,x{b}^{3}{d}^{2}e+2\,{e}^{3}{a}^{3}+12\,d{e}^{2}{a}^{2}b-48\,a{d}^{2}e{b}^{2}+32\,{d}^{3}{b}^{3}}{3\,{e}^{4}} \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)/(e*x+d)^(5/2),x)

[Out]

-2/3*(-b^3*e^3*x^3-9*a*b^2*e^3*x^2+6*b^3*d*e^2*x^2+9*a^2*b*e^3*x-36*a*b^2*d*e^2*x+24*b^3*d^2*e*x+a^3*e^3+6*a^2
*b*d*e^2-24*a*b^2*d^2*e+16*b^3*d^3)/(e*x+d)^(3/2)/e^4

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Maxima [A]  time = 0.958641, size = 165, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} b^{3} - 9 \,{\left (b^{3} d - a b^{2} e\right )} \sqrt{e x + d}}{e^{3}} + \frac{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3} - 9 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, 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)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/3*(((e*x + d)^(3/2)*b^3 - 9*(b^3*d - a*b^2*e)*sqrt(e*x + d))/e^3 + (b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2
- a^3*e^3 - 9*(b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*(e*x + d))/((e*x + d)^(3/2)*e^3))/e

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Fricas [A]  time = 1.32959, size = 281, normalized size = 2.93 \begin{align*} \frac{2 \,{\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\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)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/3*(b^3*e^3*x^3 - 16*b^3*d^3 + 24*a*b^2*d^2*e - 6*a^2*b*d*e^2 - a^3*e^3 - 3*(2*b^3*d*e^2 - 3*a*b^2*e^3)*x^2 -
3*(8*b^3*d^2*e - 12*a*b^2*d*e^2 + 3*a^2*b*e^3)*x)*sqrt(e*x + d)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

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Sympy [A]  time = 1.50962, size = 461, normalized size = 4.8 \begin{align*} \begin{cases} - \frac{2 a^{3} e^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 a^{2} b d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{18 a^{2} b e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{48 a b^{2} d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{72 a b^{2} d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{18 a b^{2} e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 b^{3} d^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 b^{3} d^{2} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 b^{3} d e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 b^{3} e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \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)/(e*x+d)**(5/2),x)

[Out]

Piecewise((-2*a**3*e**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 12*a**2*b*d*e**2/(3*d*e**4*sqrt(d
+ e*x) + 3*e**5*x*sqrt(d + e*x)) - 18*a**2*b*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 48*a*b
**2*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 72*a*b**2*d*e**2*x/(3*d*e**4*sqrt(d + e*x) + 3*
e**5*x*sqrt(d + e*x)) + 18*a*b**2*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 32*b**3*d**3/(
3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 48*b**3*d**2*e*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d
+ e*x)) - 12*b**3*d*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 2*b**3*e**3*x**3/(3*d*e**4*s
qrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), ((a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4)/d**
(5/2), True))

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Giac [A]  time = 1.14145, size = 192, normalized size = 2. \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{8} - 9 \, \sqrt{x e + d} b^{3} d e^{8} + 9 \, \sqrt{x e + d} a b^{2} e^{9}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} b^{3} d^{2} - b^{3} d^{3} - 18 \,{\left (x e + d\right )} a b^{2} d e + 3 \, a b^{2} d^{2} e + 9 \,{\left (x e + d\right )} a^{2} b e^{2} - 3 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} e^{\left (-4\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)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*b^3*e^8 - 9*sqrt(x*e + d)*b^3*d*e^8 + 9*sqrt(x*e + d)*a*b^2*e^9)*e^(-12) - 2/3*(9*(x*e +
d)*b^3*d^2 - b^3*d^3 - 18*(x*e + d)*a*b^2*d*e + 3*a*b^2*d^2*e + 9*(x*e + d)*a^2*b*e^2 - 3*a^2*b*d*e^2 + a^3*e^
3)*e^(-4)/(x*e + d)^(3/2)