3.2055 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{10 b^4 (d+e x)^{7/2} (b d-a e)}{7 e^6}+\frac{4 b^3 (d+e x)^{5/2} (b d-a e)^2}{e^6}-\frac{20 b^2 (d+e x)^{3/2} (b d-a e)^3}{3 e^6}+\frac{10 b \sqrt{d+e x} (b d-a e)^4}{e^6}+\frac{2 (b d-a e)^5}{e^6 \sqrt{d+e x}}+\frac{2 b^5 (d+e x)^{9/2}}{9 e^6} \]

[Out]

(2*(b*d - a*e)^5)/(e^6*Sqrt[d + e*x]) + (10*b*(b*d - a*e)^4*Sqrt[d + e*x])/e^6 - (20*b^2*(b*d - a*e)^3*(d + e*
x)^(3/2))/(3*e^6) + (4*b^3*(b*d - a*e)^2*(d + e*x)^(5/2))/e^6 - (10*b^4*(b*d - a*e)*(d + e*x)^(7/2))/(7*e^6) +
(2*b^5*(d + e*x)^(9/2))/(9*e^6)

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Rubi [A]  time = 0.0581177, 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} \[ -\frac{10 b^4 (d+e x)^{7/2} (b d-a e)}{7 e^6}+\frac{4 b^3 (d+e x)^{5/2} (b d-a e)^2}{e^6}-\frac{20 b^2 (d+e x)^{3/2} (b d-a e)^3}{3 e^6}+\frac{10 b \sqrt{d+e x} (b d-a e)^4}{e^6}+\frac{2 (b d-a e)^5}{e^6 \sqrt{d+e x}}+\frac{2 b^5 (d+e x)^{9/2}}{9 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0947648, size = 123, normalized size = 0.81 \[ \frac{2 \left (-210 b^2 (d+e x)^2 (b d-a e)^3+126 b^3 (d+e x)^3 (b d-a e)^2-45 b^4 (d+e x)^4 (b d-a e)+315 b (d+e x) (b d-a e)^4+63 (b d-a e)^5+7 b^5 (d+e x)^5\right )}{63 e^6 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(63*(b*d - a*e)^5 + 315*b*(b*d - a*e)^4*(d + e*x) - 210*b^2*(b*d - a*e)^3*(d + e*x)^2 + 126*b^3*(b*d - a*e)
^2*(d + e*x)^3 - 45*b^4*(b*d - a*e)*(d + e*x)^4 + 7*b^5*(d + e*x)^5))/(63*e^6*Sqrt[d + e*x])

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Maple [B]  time = 0.007, size = 273, normalized size = 1.8 \begin{align*} -{\frac{-14\,{x}^{5}{b}^{5}{e}^{5}-90\,{x}^{4}a{b}^{4}{e}^{5}+20\,{x}^{4}{b}^{5}d{e}^{4}-252\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+144\,{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}+504\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-288\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+64\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-630\,x{a}^{4}b{e}^{5}+1680\,x{a}^{3}{b}^{2}d{e}^{4}-2016\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+1152\,xa{b}^{4}{d}^{3}{e}^{2}-256\,x{b}^{5}{d}^{4}e+126\,{a}^{5}{e}^{5}-1260\,{a}^{4}bd{e}^{4}+3360\,{a}^{3}{d}^{2}{b}^{2}{e}^{3}-4032\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+2304\,a{d}^{4}{b}^{4}e-512\,{b}^{5}{d}^{5}}{63\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \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)^(3/2),x)

[Out]

-2/63*(-7*b^5*e^5*x^5-45*a*b^4*e^5*x^4+10*b^5*d*e^4*x^4-126*a^2*b^3*e^5*x^3+72*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+252*a^2*b^3*d*e^4*x^2-144*a*b^4*d^2*e^3*x^2+32*b^5*d^3*e^2*x^2-315*a^4*b*e^5*x+840*a^3
*b^2*d*e^4*x-1008*a^2*b^3*d^2*e^3*x+576*a*b^4*d^3*e^2*x-128*b^5*d^4*e*x+63*a^5*e^5-630*a^4*b*d*e^4+1680*a^3*b^
2*d^2*e^3-2016*a^2*b^3*d^3*e^2+1152*a*b^4*d^4*e-256*b^5*d^5)/(e*x+d)^(1/2)/e^6

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Maxima [A]  time = 0.98209, size = 360, normalized size = 2.37 \begin{align*} \frac{2 \,{\left (\frac{7 \,{\left (e x + d\right )}^{\frac{9}{2}} b^{5} - 45 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 126 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{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 )}{\left (e x + d\right )}^{\frac{3}{2}} + 315 \,{\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 )} \sqrt{e x + d}}{e^{5}} + \frac{63 \,{\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 )}}{\sqrt{e x + d} e^{5}}\right )}}{63 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/63*((7*(e*x + d)^(9/2)*b^5 - 45*(b^5*d - a*b^4*e)*(e*x + d)^(7/2) + 126*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2
)*(e*x + d)^(5/2) - 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)*(e*x + d)^(3/2) + 315*(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)*sqrt(e*x + d))/e^5 + 63*(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)/(sqrt(e*x + d)*e^5))/e

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Fricas [B]  time = 1.29703, size = 590, normalized size = 3.88 \begin{align*} \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \,{\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{63 \,{\left (e^{7} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b
*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*
x^3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 576*a
*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^7*x + d*e^6)

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Sympy [A]  time = 34.2543, size = 243, normalized size = 1.6 \begin{align*} \frac{2 b^{5} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (10 a b^{4} e - 10 b^{5} d\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (20 a^{2} b^{3} e^{2} - 40 a b^{4} d e + 20 b^{5} d^{2}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \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 )}{3 e^{6}} + \frac{\sqrt{d + e x} \left (10 a^{4} b e^{4} - 40 a^{3} b^{2} d e^{3} + 60 a^{2} b^{3} d^{2} e^{2} - 40 a b^{4} d^{3} e + 10 b^{5} d^{4}\right )}{e^{6}} - \frac{2 \left (a e - b d\right )^{5}}{e^{6} \sqrt{d + e x}} \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)**(3/2),x)

[Out]

2*b**5*(d + e*x)**(9/2)/(9*e**6) + (d + e*x)**(7/2)*(10*a*b**4*e - 10*b**5*d)/(7*e**6) + (d + e*x)**(5/2)*(20*
a**2*b**3*e**2 - 40*a*b**4*d*e + 20*b**5*d**2)/(5*e**6) + (d + e*x)**(3/2)*(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)/(3*e**6) + sqrt(d + e*x)*(10*a**4*b*e**4 - 40*a**3*b**2*d*e**3 + 60*a
**2*b**3*d**2*e**2 - 40*a*b**4*d**3*e + 10*b**5*d**4)/e**6 - 2*(a*e - b*d)**5/(e**6*sqrt(d + e*x))

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Giac [B]  time = 1.13722, size = 467, normalized size = 3.07 \begin{align*} \frac{2}{63} \,{\left (7 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{5} e^{48} - 45 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d e^{48} + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{2} e^{48} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{3} e^{48} + 315 \, \sqrt{x e + d} b^{5} d^{4} e^{48} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} e^{49} - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d e^{49} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{2} e^{49} - 1260 \, \sqrt{x e + d} a b^{4} d^{3} e^{49} + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} e^{50} - 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d e^{50} + 1890 \, \sqrt{x e + d} a^{2} b^{3} d^{2} e^{50} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} e^{51} - 1260 \, \sqrt{x e + d} a^{3} b^{2} d e^{51} + 315 \, \sqrt{x e + d} a^{4} b e^{52}\right )} e^{\left (-54\right )} + \frac{2 \,{\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 )} e^{\left (-6\right )}}{\sqrt{x e + d}} \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)^(3/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*b^5*e^48 - 45*(x*e + d)^(7/2)*b^5*d*e^48 + 126*(x*e + d)^(5/2)*b^5*d^2*e^48 - 210*(x*e
+ d)^(3/2)*b^5*d^3*e^48 + 315*sqrt(x*e + d)*b^5*d^4*e^48 + 45*(x*e + d)^(7/2)*a*b^4*e^49 - 252*(x*e + d)^(5/2
)*a*b^4*d*e^49 + 630*(x*e + d)^(3/2)*a*b^4*d^2*e^49 - 1260*sqrt(x*e + d)*a*b^4*d^3*e^49 + 126*(x*e + d)^(5/2)*
a^2*b^3*e^50 - 630*(x*e + d)^(3/2)*a^2*b^3*d*e^50 + 1890*sqrt(x*e + d)*a^2*b^3*d^2*e^50 + 210*(x*e + d)^(3/2)*
a^3*b^2*e^51 - 1260*sqrt(x*e + d)*a^3*b^2*d*e^51 + 315*sqrt(x*e + d)*a^4*b*e^52)*e^(-54) + 2*(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^(-6)/sqrt(x*e + d)