3.1643 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{12 b^5 (d+e x)^{7/2} (b d-a e)}{7 e^7}+\frac{6 b^4 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{3/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 \sqrt{d+e x} (b d-a e)^4}{e^7}+\frac{12 b (b d-a e)^5}{e^7 \sqrt{d+e x}}-\frac{2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac{2 b^6 (d+e x)^{9/2}}{9 e^7} \]

[Out]

(-2*(b*d - a*e)^6)/(3*e^7*(d + e*x)^(3/2)) + (12*b*(b*d - a*e)^5)/(e^7*Sqrt[d + e*x]) + (30*b^2*(b*d - a*e)^4*
Sqrt[d + e*x])/e^7 - (40*b^3*(b*d - a*e)^3*(d + e*x)^(3/2))/(3*e^7) + (6*b^4*(b*d - a*e)^2*(d + e*x)^(5/2))/e^
7 - (12*b^5*(b*d - a*e)*(d + e*x)^(7/2))/(7*e^7) + (2*b^6*(d + e*x)^(9/2))/(9*e^7)

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Rubi [A]  time = 0.0592892, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {27, 43} \[ -\frac{12 b^5 (d+e x)^{7/2} (b d-a e)}{7 e^7}+\frac{6 b^4 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{3/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 \sqrt{d+e x} (b d-a e)^4}{e^7}+\frac{12 b (b d-a e)^5}{e^7 \sqrt{d+e x}}-\frac{2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac{2 b^6 (d+e x)^{9/2}}{9 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(-2*(b*d - a*e)^6)/(3*e^7*(d + e*x)^(3/2)) + (12*b*(b*d - a*e)^5)/(e^7*Sqrt[d + e*x]) + (30*b^2*(b*d - a*e)^4*
Sqrt[d + e*x])/e^7 - (40*b^3*(b*d - a*e)^3*(d + e*x)^(3/2))/(3*e^7) + (6*b^4*(b*d - a*e)^2*(d + e*x)^(5/2))/e^
7 - (12*b^5*(b*d - a*e)*(d + e*x)^(7/2))/(7*e^7) + (2*b^6*(d + e*x)^(9/2))/(9*e^7)

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{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{5/2}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{3/2}}+\frac{15 b^2 (b d-a e)^4}{e^6 \sqrt{d+e x}}-\frac{20 b^3 (b d-a e)^3 \sqrt{d+e x}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{5/2}}{e^6}+\frac{b^6 (d+e x)^{7/2}}{e^6}\right ) \, dx\\ &=-\frac{2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac{12 b (b d-a e)^5}{e^7 \sqrt{d+e x}}+\frac{30 b^2 (b d-a e)^4 \sqrt{d+e x}}{e^7}-\frac{40 b^3 (b d-a e)^3 (d+e x)^{3/2}}{3 e^7}+\frac{6 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^7}-\frac{12 b^5 (b d-a e) (d+e x)^{7/2}}{7 e^7}+\frac{2 b^6 (d+e x)^{9/2}}{9 e^7}\\ \end{align*}

Mathematica [A]  time = 0.0834247, size = 145, normalized size = 0.8 \[ \frac{2 \left (945 b^2 (d+e x)^2 (b d-a e)^4-420 b^3 (d+e x)^3 (b d-a e)^3+189 b^4 (d+e x)^4 (b d-a e)^2-54 b^5 (d+e x)^5 (b d-a e)+378 b (d+e x) (b d-a e)^5-21 (b d-a e)^6+7 b^6 (d+e x)^6\right )}{63 e^7 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(2*(-21*(b*d - a*e)^6 + 378*b*(b*d - a*e)^5*(d + e*x) + 945*b^2*(b*d - a*e)^4*(d + e*x)^2 - 420*b^3*(b*d - a*e
)^3*(d + e*x)^3 + 189*b^4*(b*d - a*e)^2*(d + e*x)^4 - 54*b^5*(b*d - a*e)*(d + e*x)^5 + 7*b^6*(d + e*x)^6))/(63
*e^7*(d + e*x)^(3/2))

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Maple [B]  time = 0.048, size = 377, normalized size = 2.1 \begin{align*} -{\frac{-14\,{b}^{6}{x}^{6}{e}^{6}-108\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-378\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+216\,{x}^{4}a{b}^{5}d{e}^{5}-48\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-840\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+1008\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-576\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+128\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-1890\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+5040\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-6048\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+3456\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-768\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+756\,x{a}^{5}b{e}^{6}-7560\,x{a}^{4}{b}^{2}d{e}^{5}+20160\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-24192\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+13824\,xa{b}^{5}{d}^{4}{e}^{2}-3072\,x{b}^{6}{d}^{5}e+42\,{a}^{6}{e}^{6}+504\,{a}^{5}bd{e}^{5}-5040\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}+13440\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}-16128\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+9216\,a{b}^{5}{d}^{5}e-2048\,{d}^{6}{b}^{6}}{63\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x)

[Out]

-2/63*(-7*b^6*e^6*x^6-54*a*b^5*e^6*x^5+12*b^6*d*e^5*x^5-189*a^2*b^4*e^6*x^4+108*a*b^5*d*e^5*x^4-24*b^6*d^2*e^4
*x^4-420*a^3*b^3*e^6*x^3+504*a^2*b^4*d*e^5*x^3-288*a*b^5*d^2*e^4*x^3+64*b^6*d^3*e^3*x^3-945*a^4*b^2*e^6*x^2+25
20*a^3*b^3*d*e^5*x^2-3024*a^2*b^4*d^2*e^4*x^2+1728*a*b^5*d^3*e^3*x^2-384*b^6*d^4*e^2*x^2+378*a^5*b*e^6*x-3780*
a^4*b^2*d*e^5*x+10080*a^3*b^3*d^2*e^4*x-12096*a^2*b^4*d^3*e^3*x+6912*a*b^5*d^4*e^2*x-1536*b^6*d^5*e*x+21*a^6*e
^6+252*a^5*b*d*e^5-2520*a^4*b^2*d^2*e^4+6720*a^3*b^3*d^3*e^3-8064*a^2*b^4*d^4*e^2+4608*a*b^5*d^5*e-1024*b^6*d^
6)/(e*x+d)^(3/2)/e^7

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Maxima [B]  time = 1.04811, size = 481, normalized size = 2.66 \begin{align*} \frac{2 \,{\left (\frac{7 \,{\left (e x + d\right )}^{\frac{9}{2}} b^{6} - 54 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 420 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 945 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \sqrt{e x + d}}{e^{6}} - \frac{21 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6} - 18 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{6}}\right )}}{63 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/63*((7*(e*x + d)^(9/2)*b^6 - 54*(b^6*d - a*b^5*e)*(e*x + d)^(7/2) + 189*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2
)*(e*x + d)^(5/2) - 420*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d)^(3/2) + 945*(b^6*d
^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*sqrt(e*x + d))/e^6 - 21*(b^6*d^6 - 6*a
*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6 - 18*(b^6*
d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*(e*x + d))/((e*x
+ d)^(3/2)*e^6))/e

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Fricas [B]  time = 1.81573, size = 822, normalized size = 4.54 \begin{align*} \frac{2 \,{\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 4608 \, a b^{5} d^{5} e + 8064 \, a^{2} b^{4} d^{4} e^{2} - 6720 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} - 252 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} - 6 \,{\left (2 \, b^{6} d e^{5} - 9 \, a b^{5} e^{6}\right )} x^{5} + 3 \,{\left (8 \, b^{6} d^{2} e^{4} - 36 \, a b^{5} d e^{5} + 63 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \,{\left (16 \, b^{6} d^{3} e^{3} - 72 \, a b^{5} d^{2} e^{4} + 126 \, a^{2} b^{4} d e^{5} - 105 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{4} e^{2} - 576 \, a b^{5} d^{3} e^{3} + 1008 \, a^{2} b^{4} d^{2} e^{4} - 840 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (256 \, b^{6} d^{5} e - 1152 \, a b^{5} d^{4} e^{2} + 2016 \, a^{2} b^{4} d^{3} e^{3} - 1680 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} - 63 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{63 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/63*(7*b^6*e^6*x^6 + 1024*b^6*d^6 - 4608*a*b^5*d^5*e + 8064*a^2*b^4*d^4*e^2 - 6720*a^3*b^3*d^3*e^3 + 2520*a^4
*b^2*d^2*e^4 - 252*a^5*b*d*e^5 - 21*a^6*e^6 - 6*(2*b^6*d*e^5 - 9*a*b^5*e^6)*x^5 + 3*(8*b^6*d^2*e^4 - 36*a*b^5*
d*e^5 + 63*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 72*a*b^5*d^2*e^4 + 126*a^2*b^4*d*e^5 - 105*a^3*b^3*e^6)*x^3
+ 3*(128*b^6*d^4*e^2 - 576*a*b^5*d^3*e^3 + 1008*a^2*b^4*d^2*e^4 - 840*a^3*b^3*d*e^5 + 315*a^4*b^2*e^6)*x^2 + 6
*(256*b^6*d^5*e - 1152*a*b^5*d^4*e^2 + 2016*a^2*b^4*d^3*e^3 - 1680*a^3*b^3*d^2*e^4 + 630*a^4*b^2*d*e^5 - 63*a^
5*b*e^6)*x)*sqrt(e*x + d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 66.3361, size = 270, normalized size = 1.49 \begin{align*} \frac{2 b^{6} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{7}} - \frac{12 b \left (a e - b d\right )^{5}}{e^{7} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (12 a b^{5} e - 12 b^{6} d\right )}{7 e^{7}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (30 a^{2} b^{4} e^{2} - 60 a b^{5} d e + 30 b^{6} d^{2}\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (40 a^{3} b^{3} e^{3} - 120 a^{2} b^{4} d e^{2} + 120 a b^{5} d^{2} e - 40 b^{6} d^{3}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (30 a^{4} b^{2} e^{4} - 120 a^{3} b^{3} d e^{3} + 180 a^{2} b^{4} d^{2} e^{2} - 120 a b^{5} d^{3} e + 30 b^{6} d^{4}\right )}{e^{7}} - \frac{2 \left (a e - b d\right )^{6}}{3 e^{7} \left (d + e x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(5/2),x)

[Out]

2*b**6*(d + e*x)**(9/2)/(9*e**7) - 12*b*(a*e - b*d)**5/(e**7*sqrt(d + e*x)) + (d + e*x)**(7/2)*(12*a*b**5*e -
12*b**6*d)/(7*e**7) + (d + e*x)**(5/2)*(30*a**2*b**4*e**2 - 60*a*b**5*d*e + 30*b**6*d**2)/(5*e**7) + (d + e*x)
**(3/2)*(40*a**3*b**3*e**3 - 120*a**2*b**4*d*e**2 + 120*a*b**5*d**2*e - 40*b**6*d**3)/(3*e**7) + sqrt(d + e*x)
*(30*a**4*b**2*e**4 - 120*a**3*b**3*d*e**3 + 180*a**2*b**4*d**2*e**2 - 120*a*b**5*d**3*e + 30*b**6*d**4)/e**7
- 2*(a*e - b*d)**6/(3*e**7*(d + e*x)**(3/2))

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Giac [B]  time = 1.22098, size = 624, normalized size = 3.45 \begin{align*} \frac{2}{63} \,{\left (7 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{6} e^{56} - 54 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} d e^{56} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d^{2} e^{56} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{3} e^{56} + 945 \, \sqrt{x e + d} b^{6} d^{4} e^{56} + 54 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{5} e^{57} - 378 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} d e^{57} + 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d^{2} e^{57} - 3780 \, \sqrt{x e + d} a b^{5} d^{3} e^{57} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{4} e^{58} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} d e^{58} + 5670 \, \sqrt{x e + d} a^{2} b^{4} d^{2} e^{58} + 420 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{3} e^{59} - 3780 \, \sqrt{x e + d} a^{3} b^{3} d e^{59} + 945 \, \sqrt{x e + d} a^{4} b^{2} e^{60}\right )} e^{\left (-63\right )} + \frac{2 \,{\left (18 \,{\left (x e + d\right )} b^{6} d^{5} - b^{6} d^{6} - 90 \,{\left (x e + d\right )} a b^{5} d^{4} e + 6 \, a b^{5} d^{5} e + 180 \,{\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} - 15 \, a^{2} b^{4} d^{4} e^{2} - 180 \,{\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{3} d^{3} e^{3} + 90 \,{\left (x e + d\right )} a^{4} b^{2} d e^{4} - 15 \, a^{4} b^{2} d^{2} e^{4} - 18 \,{\left (x e + d\right )} a^{5} b e^{5} + 6 \, a^{5} b d e^{5} - a^{6} e^{6}\right )} e^{\left (-7\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^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*b^6*e^56 - 54*(x*e + d)^(7/2)*b^6*d*e^56 + 189*(x*e + d)^(5/2)*b^6*d^2*e^56 - 420*(x*e
+ d)^(3/2)*b^6*d^3*e^56 + 945*sqrt(x*e + d)*b^6*d^4*e^56 + 54*(x*e + d)^(7/2)*a*b^5*e^57 - 378*(x*e + d)^(5/2
)*a*b^5*d*e^57 + 1260*(x*e + d)^(3/2)*a*b^5*d^2*e^57 - 3780*sqrt(x*e + d)*a*b^5*d^3*e^57 + 189*(x*e + d)^(5/2)
*a^2*b^4*e^58 - 1260*(x*e + d)^(3/2)*a^2*b^4*d*e^58 + 5670*sqrt(x*e + d)*a^2*b^4*d^2*e^58 + 420*(x*e + d)^(3/2
)*a^3*b^3*e^59 - 3780*sqrt(x*e + d)*a^3*b^3*d*e^59 + 945*sqrt(x*e + d)*a^4*b^2*e^60)*e^(-63) + 2/3*(18*(x*e +
d)*b^6*d^5 - b^6*d^6 - 90*(x*e + d)*a*b^5*d^4*e + 6*a*b^5*d^5*e + 180*(x*e + d)*a^2*b^4*d^3*e^2 - 15*a^2*b^4*d
^4*e^2 - 180*(x*e + d)*a^3*b^3*d^2*e^3 + 20*a^3*b^3*d^3*e^3 + 90*(x*e + d)*a^4*b^2*d*e^4 - 15*a^4*b^2*d^2*e^4
- 18*(x*e + d)*a^5*b*e^5 + 6*a^5*b*d*e^5 - a^6*e^6)*e^(-7)/(x*e + d)^(3/2)