3.1644 $$\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=179 $-\frac{12 b^5 (d+e x)^{5/2} (b d-a e)}{5 e^7}+\frac{10 b^4 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac{40 b^3 \sqrt{d+e x} (b d-a e)^3}{e^7}-\frac{30 b^2 (b d-a e)^4}{e^7 \sqrt{d+e x}}+\frac{4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac{2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac{2 b^6 (d+e x)^{7/2}}{7 e^7}$

[Out]

(-2*(b*d - a*e)^6)/(5*e^7*(d + e*x)^(5/2)) + (4*b*(b*d - a*e)^5)/(e^7*(d + e*x)^(3/2)) - (30*b^2*(b*d - a*e)^4
)/(e^7*Sqrt[d + e*x]) - (40*b^3*(b*d - a*e)^3*Sqrt[d + e*x])/e^7 + (10*b^4*(b*d - a*e)^2*(d + e*x)^(3/2))/e^7
- (12*b^5*(b*d - a*e)*(d + e*x)^(5/2))/(5*e^7) + (2*b^6*(d + e*x)^(7/2))/(7*e^7)

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Rubi [A]  time = 0.0591365, antiderivative size = 179, 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} $-\frac{12 b^5 (d+e x)^{5/2} (b d-a e)}{5 e^7}+\frac{10 b^4 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac{40 b^3 \sqrt{d+e x} (b d-a e)^3}{e^7}-\frac{30 b^2 (b d-a e)^4}{e^7 \sqrt{d+e x}}+\frac{4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac{2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac{2 b^6 (d+e x)^{7/2}}{7 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^6)/(5*e^7*(d + e*x)^(5/2)) + (4*b*(b*d - a*e)^5)/(e^7*(d + e*x)^(3/2)) - (30*b^2*(b*d - a*e)^4
)/(e^7*Sqrt[d + e*x]) - (40*b^3*(b*d - a*e)^3*Sqrt[d + e*x])/e^7 + (10*b^4*(b*d - a*e)^2*(d + e*x)^(3/2))/e^7
- (12*b^5*(b*d - a*e)*(d + e*x)^(5/2))/(5*e^7) + (2*b^6*(d + e*x)^(7/2))/(7*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)^{7/2}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{7/2}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{5/2}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{3/2}}-\frac{20 b^3 (b d-a e)^3}{e^6 \sqrt{d+e x}}+\frac{15 b^4 (b d-a e)^2 \sqrt{d+e x}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{3/2}}{e^6}+\frac{b^6 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac{2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac{4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac{30 b^2 (b d-a e)^4}{e^7 \sqrt{d+e x}}-\frac{40 b^3 (b d-a e)^3 \sqrt{d+e x}}{e^7}+\frac{10 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^7}-\frac{12 b^5 (b d-a e) (d+e x)^{5/2}}{5 e^7}+\frac{2 b^6 (d+e x)^{7/2}}{7 e^7}\\ \end{align*}

Mathematica [A]  time = 0.0814156, size = 145, normalized size = 0.81 $\frac{2 \left (-525 b^2 (d+e x)^2 (b d-a e)^4-700 b^3 (d+e x)^3 (b d-a e)^3+175 b^4 (d+e x)^4 (b d-a e)^2-42 b^5 (d+e x)^5 (b d-a e)+70 b (d+e x) (b d-a e)^5-7 (b d-a e)^6+5 b^6 (d+e x)^6\right )}{35 e^7 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(2*(-7*(b*d - a*e)^6 + 70*b*(b*d - a*e)^5*(d + e*x) - 525*b^2*(b*d - a*e)^4*(d + e*x)^2 - 700*b^3*(b*d - a*e)^
3*(d + e*x)^3 + 175*b^4*(b*d - a*e)^2*(d + e*x)^4 - 42*b^5*(b*d - a*e)*(d + e*x)^5 + 5*b^6*(d + e*x)^6))/(35*e
^7*(d + e*x)^(5/2))

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Maple [B]  time = 0.048, size = 377, normalized size = 2.1 \begin{align*} -{\frac{-10\,{b}^{6}{x}^{6}{e}^{6}-84\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-350\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+280\,{x}^{4}a{b}^{5}d{e}^{5}-80\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-1400\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+2800\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-2240\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+640\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+1050\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-8400\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+16800\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-13440\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+3840\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+140\,x{a}^{5}b{e}^{6}+1400\,x{a}^{4}{b}^{2}d{e}^{5}-11200\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+22400\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-17920\,xa{b}^{5}{d}^{4}{e}^{2}+5120\,x{b}^{6}{d}^{5}e+14\,{a}^{6}{e}^{6}+56\,{a}^{5}bd{e}^{5}+560\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-4480\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}+8960\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-7168\,a{b}^{5}{d}^{5}e+2048\,{d}^{6}{b}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{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)^(7/2),x)

[Out]

-2/35*(-5*b^6*e^6*x^6-42*a*b^5*e^6*x^5+12*b^6*d*e^5*x^5-175*a^2*b^4*e^6*x^4+140*a*b^5*d*e^5*x^4-40*b^6*d^2*e^4
*x^4-700*a^3*b^3*e^6*x^3+1400*a^2*b^4*d*e^5*x^3-1120*a*b^5*d^2*e^4*x^3+320*b^6*d^3*e^3*x^3+525*a^4*b^2*e^6*x^2
-4200*a^3*b^3*d*e^5*x^2+8400*a^2*b^4*d^2*e^4*x^2-6720*a*b^5*d^3*e^3*x^2+1920*b^6*d^4*e^2*x^2+70*a^5*b*e^6*x+70
0*a^4*b^2*d*e^5*x-5600*a^3*b^3*d^2*e^4*x+11200*a^2*b^4*d^3*e^3*x-8960*a*b^5*d^4*e^2*x+2560*b^6*d^5*e*x+7*a^6*e
^6+28*a^5*b*d*e^5+280*a^4*b^2*d^2*e^4-2240*a^3*b^3*d^3*e^3+4480*a^2*b^4*d^4*e^2-3584*a*b^5*d^5*e+1024*b^6*d^6)
/(e*x+d)^(5/2)/e^7

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Maxima [B]  time = 1.17603, size = 481, normalized size = 2.69 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{6} - 42 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 175 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 700 \,{\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 )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\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} + 75 \,{\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 )}{\left (e x + d\right )}^{2} - 10 \,{\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{5}{2}} e^{6}}\right )}}{35 \, 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)^(7/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*b^6 - 42*(b^6*d - a*b^5*e)*(e*x + d)^(5/2) + 175*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2
)*(e*x + d)^(3/2) - 700*(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)*sqrt(e*x + d))/e^6 - 7*(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 +
75*(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)*(e*x + d)^2 - 10*(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)^(
5/2)*e^6))/e

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Fricas [B]  time = 1.84991, size = 834, normalized size = 4.66 \begin{align*} \frac{2 \,{\left (5 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 3584 \, a b^{5} d^{5} e - 4480 \, a^{2} b^{4} d^{4} e^{2} + 2240 \, a^{3} b^{3} d^{3} e^{3} - 280 \, a^{4} b^{2} d^{2} e^{4} - 28 \, a^{5} b d e^{5} - 7 \, a^{6} e^{6} - 6 \,{\left (2 \, b^{6} d e^{5} - 7 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (8 \, b^{6} d^{2} e^{4} - 28 \, a b^{5} d e^{5} + 35 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (16 \, b^{6} d^{3} e^{3} - 56 \, a b^{5} d^{2} e^{4} + 70 \, a^{2} b^{4} d e^{5} - 35 \, a^{3} b^{3} e^{6}\right )} x^{3} - 15 \,{\left (128 \, b^{6} d^{4} e^{2} - 448 \, a b^{5} d^{3} e^{3} + 560 \, a^{2} b^{4} d^{2} e^{4} - 280 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} - 10 \,{\left (256 \, b^{6} d^{5} e - 896 \, a b^{5} d^{4} e^{2} + 1120 \, a^{2} b^{4} d^{3} e^{3} - 560 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 7 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*b^6*e^6*x^6 - 1024*b^6*d^6 + 3584*a*b^5*d^5*e - 4480*a^2*b^4*d^4*e^2 + 2240*a^3*b^3*d^3*e^3 - 280*a^4*
b^2*d^2*e^4 - 28*a^5*b*d*e^5 - 7*a^6*e^6 - 6*(2*b^6*d*e^5 - 7*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 28*a*b^5*d*e
^5 + 35*a^2*b^4*e^6)*x^4 - 20*(16*b^6*d^3*e^3 - 56*a*b^5*d^2*e^4 + 70*a^2*b^4*d*e^5 - 35*a^3*b^3*e^6)*x^3 - 15
*(128*b^6*d^4*e^2 - 448*a*b^5*d^3*e^3 + 560*a^2*b^4*d^2*e^4 - 280*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 - 10*(25
6*b^6*d^5*e - 896*a*b^5*d^4*e^2 + 1120*a^2*b^4*d^3*e^3 - 560*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 7*a^5*b*e^6)
*x)*sqrt(e*x + d)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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Sympy [A]  time = 123.569, size = 221, normalized size = 1.23 \begin{align*} \frac{2 b^{6} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} - \frac{30 b^{2} \left (a e - b d\right )^{4}}{e^{7} \sqrt{d + e x}} - \frac{4 b \left (a e - b d\right )^{5}}{e^{7} \left (d + e x\right )^{\frac{3}{2}}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (12 a b^{5} e - 12 b^{6} d\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (30 a^{2} b^{4} e^{2} - 60 a b^{5} d e + 30 b^{6} d^{2}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \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 )}{e^{7}} - \frac{2 \left (a e - b d\right )^{6}}{5 e^{7} \left (d + e x\right )^{\frac{5}{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)**(7/2),x)

[Out]

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

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Giac [B]  time = 1.23942, size = 618, normalized size = 3.45 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{2} e^{42} - 700 \, \sqrt{x e + d} b^{6} d^{3} e^{42} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} e^{43} - 350 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d e^{43} + 2100 \, \sqrt{x e + d} a b^{5} d^{2} e^{43} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} e^{44} - 2100 \, \sqrt{x e + d} a^{2} b^{4} d e^{44} + 700 \, \sqrt{x e + d} a^{3} b^{3} e^{45}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} b^{6} d^{4} - 10 \,{\left (x e + d\right )} b^{6} d^{5} + b^{6} d^{6} - 300 \,{\left (x e + d\right )}^{2} a b^{5} d^{3} e + 50 \,{\left (x e + d\right )} a b^{5} d^{4} e - 6 \, a b^{5} d^{5} e + 450 \,{\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} - 100 \,{\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{2} - 300 \,{\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} + 100 \,{\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{3} + 75 \,{\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} - 50 \,{\left (x e + d\right )} a^{4} b^{2} d e^{4} + 15 \, a^{4} b^{2} d^{2} e^{4} + 10 \,{\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 )}}{5 \,{\left (x e + d\right )}^{\frac{5}{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)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*b^6*e^42 - 42*(x*e + d)^(5/2)*b^6*d*e^42 + 175*(x*e + d)^(3/2)*b^6*d^2*e^42 - 700*sqrt
(x*e + d)*b^6*d^3*e^42 + 42*(x*e + d)^(5/2)*a*b^5*e^43 - 350*(x*e + d)^(3/2)*a*b^5*d*e^43 + 2100*sqrt(x*e + d)
*a*b^5*d^2*e^43 + 175*(x*e + d)^(3/2)*a^2*b^4*e^44 - 2100*sqrt(x*e + d)*a^2*b^4*d*e^44 + 700*sqrt(x*e + d)*a^3
*b^3*e^45)*e^(-49) - 2/5*(75*(x*e + d)^2*b^6*d^4 - 10*(x*e + d)*b^6*d^5 + b^6*d^6 - 300*(x*e + d)^2*a*b^5*d^3*
e + 50*(x*e + d)*a*b^5*d^4*e - 6*a*b^5*d^5*e + 450*(x*e + d)^2*a^2*b^4*d^2*e^2 - 100*(x*e + d)*a^2*b^4*d^3*e^2
+ 15*a^2*b^4*d^4*e^2 - 300*(x*e + d)^2*a^3*b^3*d*e^3 + 100*(x*e + d)*a^3*b^3*d^2*e^3 - 20*a^3*b^3*d^3*e^3 + 7
5*(x*e + d)^2*a^4*b^2*e^4 - 50*(x*e + d)*a^4*b^2*d*e^4 + 15*a^4*b^2*d^2*e^4 + 10*(x*e + d)*a^5*b*e^5 - 6*a^5*b
*d*e^5 + a^6*e^6)*e^(-7)/(x*e + d)^(5/2)