3.1697 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=314 \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^6 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{3 e^6 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^6 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0969155, antiderivative size = 314, 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} \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^6 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{3 e^6 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^6 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^4*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*Sqrt[d + e*x]) - (20*b^2*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(e^6*(a + b*x)) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a
+ b*x)) - (2*b^4*(b*d - a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) + (2*b^5*(d + e*x
)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*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)^{5/2}} \, 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)^{5/2}} \, 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)^{5/2}}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^{3/2}}-\frac{10 b^7 (b d-a e)^3}{e^5 \sqrt{d+e x}}+\frac{10 b^8 (b d-a e)^2 \sqrt{d+e x}}{e^5}-\frac{5 b^9 (b d-a e) (d+e x)^{3/2}}{e^5}+\frac{b^{10} (d+e x)^{5/2}}{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}}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{10 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}-\frac{20 b^2 (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac{20 b^3 (b d-a e)^2 (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^4 (b d-a e) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac{2 b^5 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.116292, size = 235, normalized size = 0.75 \[ -\frac{2 \sqrt{(a+b x)^2} \left (70 a^2 b^3 e^2 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )-70 a^3 b^2 e^3 \left (8 d^2+12 d e x+3 e^2 x^2\right )+35 a^4 b e^4 (2 d+3 e x)+7 a^5 e^5-7 a b^4 e \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )+b^5 \left (96 d^3 e^2 x^2-16 d^2 e^3 x^3+384 d^4 e x+256 d^5+6 d e^4 x^4-3 e^5 x^5\right )\right )}{21 e^6 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(7*a^5*e^5 + 35*a^4*b*e^4*(2*d + 3*e*x) - 70*a^3*b^2*e^3*(8*d^2 + 12*d*e*x + 3*e^2*x^2)
+ 70*a^2*b^3*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3) - 7*a*b^4*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2
*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + b^5*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 -
3*e^5*x^5)))/(21*e^6*(a + b*x)*(d + e*x)^(3/2))

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Maple [A]  time = 0.153, size = 289, normalized size = 0.9 \begin{align*} -{\frac{-6\,{x}^{5}{b}^{5}{e}^{5}-42\,{x}^{4}a{b}^{4}{e}^{5}+12\,{x}^{4}{b}^{5}d{e}^{4}-140\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+112\,{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}+840\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-672\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+192\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+210\,x{a}^{4}b{e}^{5}-1680\,x{a}^{3}{b}^{2}d{e}^{4}+3360\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-2688\,xa{b}^{4}{d}^{3}{e}^{2}+768\,x{b}^{5}{d}^{4}e+14\,{a}^{5}{e}^{5}+140\,d{e}^{4}{a}^{4}b-1120\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+2240\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-1792\,a{b}^{4}{d}^{4}e+512\,{b}^{5}{d}^{5}}{21\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \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)^(5/2)/(e*x+d)^(5/2),x)

[Out]

-2/21/(e*x+d)^(3/2)*(-3*b^5*e^5*x^5-21*a*b^4*e^5*x^4+6*b^5*d*e^4*x^4-70*a^2*b^3*e^5*x^3+56*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+420*a^2*b^3*d*e^4*x^2-336*a*b^4*d^2*e^3*x^2+96*b^5*d^3*e^2*x^2+105*a^4*b*e
^5*x-840*a^3*b^2*d*e^4*x+1680*a^2*b^3*d^2*e^3*x-1344*a*b^4*d^3*e^2*x+384*b^5*d^4*e*x+7*a^5*e^5+70*a^4*b*d*e^4-
560*a^3*b^2*d^2*e^3+1120*a^2*b^3*d^3*e^2-896*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 = 1.13879, size = 367, normalized size = 1.17 \begin{align*} \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \,{\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{21 \,{\left (e^{7} x + d 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)^(5/2),x, algorithm="maxima")

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*
e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3
- 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e - 448*a*b^4
*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/((e^7*x + d*e^6)*sqrt(e*x + d))

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Fricas [A]  time = 1.51022, size = 603, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \,{\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{21 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*
e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3
- 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e - 448*a*b^4
*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d^2
*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)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.21762, size = 621, normalized size = 1.98 \begin{align*} \frac{2}{21} \,{\left (3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} e^{36} \mathrm{sgn}\left (b x + a\right ) - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d e^{36} \mathrm{sgn}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{2} e^{36} \mathrm{sgn}\left (b x + a\right ) - 210 \, \sqrt{x e + d} b^{5} d^{3} e^{36} \mathrm{sgn}\left (b x + a\right ) + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} e^{37} \mathrm{sgn}\left (b x + a\right ) - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d e^{37} \mathrm{sgn}\left (b x + a\right ) + 630 \, \sqrt{x e + d} a b^{4} d^{2} e^{37} \mathrm{sgn}\left (b x + a\right ) + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} e^{38} \mathrm{sgn}\left (b x + a\right ) - 630 \, \sqrt{x e + d} a^{2} b^{3} d e^{38} \mathrm{sgn}\left (b x + a\right ) + 210 \, \sqrt{x e + d} a^{3} b^{2} e^{39} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-42\right )} - \frac{2 \,{\left (15 \,{\left (x e + d\right )} b^{5} d^{4} \mathrm{sgn}\left (b x + a\right ) - b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 60 \,{\left (x e + d\right )} a b^{4} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 5 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 90 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 60 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \,{\left (x e + d\right )} a^{4} b e^{4} \mathrm{sgn}\left (b x + a\right ) - 5 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\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)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/21*(3*(x*e + d)^(7/2)*b^5*e^36*sgn(b*x + a) - 21*(x*e + d)^(5/2)*b^5*d*e^36*sgn(b*x + a) + 70*(x*e + d)^(3/2
)*b^5*d^2*e^36*sgn(b*x + a) - 210*sqrt(x*e + d)*b^5*d^3*e^36*sgn(b*x + a) + 21*(x*e + d)^(5/2)*a*b^4*e^37*sgn(
b*x + a) - 140*(x*e + d)^(3/2)*a*b^4*d*e^37*sgn(b*x + a) + 630*sqrt(x*e + d)*a*b^4*d^2*e^37*sgn(b*x + a) + 70*
(x*e + d)^(3/2)*a^2*b^3*e^38*sgn(b*x + a) - 630*sqrt(x*e + d)*a^2*b^3*d*e^38*sgn(b*x + a) + 210*sqrt(x*e + d)*
a^3*b^2*e^39*sgn(b*x + a))*e^(-42) - 2/3*(15*(x*e + d)*b^5*d^4*sgn(b*x + a) - b^5*d^5*sgn(b*x + a) - 60*(x*e +
d)*a*b^4*d^3*e*sgn(b*x + a) + 5*a*b^4*d^4*e*sgn(b*x + a) + 90*(x*e + d)*a^2*b^3*d^2*e^2*sgn(b*x + a) - 10*a^2
*b^3*d^3*e^2*sgn(b*x + a) - 60*(x*e + d)*a^3*b^2*d*e^3*sgn(b*x + a) + 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 15*(x*
e + d)*a^4*b*e^4*sgn(b*x + a) - 5*a^4*b*d*e^4*sgn(b*x + a) + a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^(3/2)