3.1698 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^6 (a+b x)}+\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) \sqrt{d+e x}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0972174, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^6 (a+b x)}+\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) \sqrt{d+e x}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(7/2),x]

[Out]

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

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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 )^{5/2}}{(d+e x)^{7/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^{7/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^5 (b d-a e)^5}{e^5 (d+e x)^{7/2}}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^{5/2}}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)^{3/2}}+\frac{10 b^8 (b d-a e)^2}{e^5 \sqrt{d+e x}}-\frac{5 b^9 (b d-a e) \sqrt{d+e x}}{e^5}+\frac{b^{10} (d+e x)^{3/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{2 (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^{5/2}}-\frac{10 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{20 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}+\frac{20 b^3 (b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac{10 b^4 (b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac{2 b^5 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.115685, size = 236, normalized size = 0.75 \[ -\frac{2 \sqrt{(a+b x)^2} \left (-30 a^2 b^3 e^2 \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+10 a^3 b^2 e^3 \left (8 d^2+20 d e x+15 e^2 x^2\right )+5 a^4 b e^4 (2 d+5 e x)+3 a^5 e^5+5 a b^4 e \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )+b^5 \left (-\left (480 d^3 e^2 x^2+80 d^2 e^3 x^3+640 d^4 e x+256 d^5-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{15 e^6 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(7/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(3*a^5*e^5 + 5*a^4*b*e^4*(2*d + 5*e*x) + 10*a^3*b^2*e^3*(8*d^2 + 20*d*e*x + 15*e^2*x^2)
- 30*a^2*b^3*e^2*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + 5*a*b^4*e*(128*d^4 + 320*d^3*e*x + 240*d^2
*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) - b^5*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^
4*x^4 + 3*e^5*x^5)))/(15*e^6*(a + b*x)*(d + e*x)^(5/2))

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Maple [A]  time = 0.154, size = 289, normalized size = 0.9 \begin{align*} -{\frac{-6\,{x}^{5}{b}^{5}{e}^{5}-50\,{x}^{4}a{b}^{4}{e}^{5}+20\,{x}^{4}{b}^{5}d{e}^{4}-300\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+400\,{x}^{3}a{b}^{4}d{e}^{4}-160\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+300\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-1800\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+2400\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-960\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+50\,x{a}^{4}b{e}^{5}+400\,x{a}^{3}{b}^{2}d{e}^{4}-2400\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+3200\,xa{b}^{4}{d}^{3}{e}^{2}-1280\,x{b}^{5}{d}^{4}e+6\,{a}^{5}{e}^{5}+20\,d{e}^{4}{a}^{4}b+160\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-960\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+1280\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{15\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \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)^(5/2)/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(-3*b^5*e^5*x^5-25*a*b^4*e^5*x^4+10*b^5*d*e^4*x^4-150*a^2*b^3*e^5*x^3+200*a*b^4*d*e^4*x^3-
80*b^5*d^2*e^3*x^3+150*a^3*b^2*e^5*x^2-900*a^2*b^3*d*e^4*x^2+1200*a*b^4*d^2*e^3*x^2-480*b^5*d^3*e^2*x^2+25*a^4
*b*e^5*x+200*a^3*b^2*d*e^4*x-1200*a^2*b^3*d^2*e^3*x+1600*a*b^4*d^3*e^2*x-640*b^5*d^4*e*x+3*a^5*e^5+10*a^4*b*d*
e^4+80*a^3*b^2*d^2*e^3-480*a^2*b^3*d^3*e^2+640*a*b^4*d^4*e-256*b^5*d^5)*((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

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Maxima [A]  time = 1.09525, size = 382, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )}}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 +
30*(16*b^5*d^3*e^2 - 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x^2 + 5*(128*b^5*d^4*e - 320*a*b^4*
d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(e*x +
d))

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Fricas [A]  time = 1.57548, size = 621, normalized size = 1.97 \begin{align*} \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\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)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 +
30*(16*b^5*d^3*e^2 - 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x^2 + 5*(128*b^5*d^4*e - 320*a*b^4*
d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^
2*e^7*x + d^3*e^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.21552, size = 620, normalized size = 1.96 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{24} \mathrm{sgn}\left (b x + a\right ) - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{24} \mathrm{sgn}\left (b x + a\right ) + 150 \, \sqrt{x e + d} b^{5} d^{2} e^{24} \mathrm{sgn}\left (b x + a\right ) + 25 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} e^{25} \mathrm{sgn}\left (b x + a\right ) - 300 \, \sqrt{x e + d} a b^{4} d e^{25} \mathrm{sgn}\left (b x + a\right ) + 150 \, \sqrt{x e + d} a^{2} b^{3} e^{26} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-30\right )} + \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} b^{5} d^{3} \mathrm{sgn}\left (b x + a\right ) - 25 \,{\left (x e + d\right )} b^{5} d^{4} \mathrm{sgn}\left (b x + a\right ) + 3 \, b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 450 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 100 \,{\left (x e + d\right )} a b^{4} d^{3} e \mathrm{sgn}\left (b x + a\right ) - 15 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 450 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm{sgn}\left (b x + a\right ) - 150 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 30 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 150 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 100 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm{sgn}\left (b x + a\right ) - 30 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 25 \,{\left (x e + d\right )} a^{4} b e^{4} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) - 3 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{15 \,{\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)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*b^5*e^24*sgn(b*x + a) - 25*(x*e + d)^(3/2)*b^5*d*e^24*sgn(b*x + a) + 150*sqrt(x*e + d)
*b^5*d^2*e^24*sgn(b*x + a) + 25*(x*e + d)^(3/2)*a*b^4*e^25*sgn(b*x + a) - 300*sqrt(x*e + d)*a*b^4*d*e^25*sgn(b
*x + a) + 150*sqrt(x*e + d)*a^2*b^3*e^26*sgn(b*x + a))*e^(-30) + 2/15*(150*(x*e + d)^2*b^5*d^3*sgn(b*x + a) -
25*(x*e + d)*b^5*d^4*sgn(b*x + a) + 3*b^5*d^5*sgn(b*x + a) - 450*(x*e + d)^2*a*b^4*d^2*e*sgn(b*x + a) + 100*(x
*e + d)*a*b^4*d^3*e*sgn(b*x + a) - 15*a*b^4*d^4*e*sgn(b*x + a) + 450*(x*e + d)^2*a^2*b^3*d*e^2*sgn(b*x + a) -
150*(x*e + d)*a^2*b^3*d^2*e^2*sgn(b*x + a) + 30*a^2*b^3*d^3*e^2*sgn(b*x + a) - 150*(x*e + d)^2*a^3*b^2*e^3*sgn
(b*x + a) + 100*(x*e + d)*a^3*b^2*d*e^3*sgn(b*x + a) - 30*a^3*b^2*d^2*e^3*sgn(b*x + a) - 25*(x*e + d)*a^4*b*e^
4*sgn(b*x + a) + 15*a^4*b*d*e^4*sgn(b*x + a) - 3*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^(5/2)