3.1696 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=314 \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^6 (a+b x)}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^6 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}{e^6 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.0950377, 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} \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^6 (a+b x)}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^6 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}{e^6 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.133589, size = 232, normalized size = 0.74 \[ \frac{2 \sqrt{(a+b x)^2} \left (126 a^2 b^3 e^2 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+210 a^3 b^2 e^3 \left (-8 d^2-4 d e x+e^2 x^2\right )+315 a^4 b e^4 (2 d+e x)-63 a^5 e^5+9 a b^4 e \left (16 d^2 e^2 x^2-64 d^3 e x-128 d^4-8 d e^3 x^3+5 e^4 x^4\right )+b^5 \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )\right )}{63 e^6 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(-63*a^5*e^5 + 315*a^4*b*e^4*(2*d + e*x) + 210*a^3*b^2*e^3*(-8*d^2 - 4*d*e*x + e^2*x^2) +
126*a^2*b^3*e^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 9*a*b^4*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*
x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + b^5*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 +
7*e^5*x^5)))/(63*e^6*(a + b*x)*Sqrt[d + e*x])

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Maple [A]  time = 0.157, size = 289, normalized size = 0.9 \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\,d{e}^{4}{a}^{4}b+3360\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-4032\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+2304\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{63\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{ex+d}}}} \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)^(3/2),x)

[Out]

-2/63/(e*x+d)^(1/2)*(-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^3-1
6*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)*((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

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Maxima [A]  time = 1.06593, size = 352, normalized size = 1.12 \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 )}}{63 \, \sqrt{e x + d} e^{6}} \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)^(3/2),x, algorithm="maxima")

[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^6)

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Fricas [A]  time = 1.63708, size = 590, normalized size = 1.88 \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^2*x^2+2*a*b*x+a^2)^(5/2)/(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 [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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.24668, size = 637, normalized size = 2.03 \begin{align*} \frac{2}{63} \,{\left (7 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{5} e^{48} \mathrm{sgn}\left (b x + a\right ) - 45 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d e^{48} \mathrm{sgn}\left (b x + a\right ) + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{2} e^{48} \mathrm{sgn}\left (b x + a\right ) - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{3} e^{48} \mathrm{sgn}\left (b x + a\right ) + 315 \, \sqrt{x e + d} b^{5} d^{4} e^{48} \mathrm{sgn}\left (b x + a\right ) + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} e^{49} \mathrm{sgn}\left (b x + a\right ) - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d e^{49} \mathrm{sgn}\left (b x + a\right ) + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{2} e^{49} \mathrm{sgn}\left (b x + a\right ) - 1260 \, \sqrt{x e + d} a b^{4} d^{3} e^{49} \mathrm{sgn}\left (b x + a\right ) + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} e^{50} \mathrm{sgn}\left (b x + a\right ) - 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d e^{50} \mathrm{sgn}\left (b x + a\right ) + 1890 \, \sqrt{x e + d} a^{2} b^{3} d^{2} e^{50} \mathrm{sgn}\left (b x + a\right ) + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} e^{51} \mathrm{sgn}\left (b x + a\right ) - 1260 \, \sqrt{x e + d} a^{3} b^{2} d e^{51} \mathrm{sgn}\left (b x + a\right ) + 315 \, \sqrt{x e + d} a^{4} b e^{52} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-54\right )} + \frac{2 \,{\left (b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \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 )}}{\sqrt{x e + 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)^(3/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*b^5*e^48*sgn(b*x + a) - 45*(x*e + d)^(7/2)*b^5*d*e^48*sgn(b*x + a) + 126*(x*e + d)^(5/
2)*b^5*d^2*e^48*sgn(b*x + a) - 210*(x*e + d)^(3/2)*b^5*d^3*e^48*sgn(b*x + a) + 315*sqrt(x*e + d)*b^5*d^4*e^48*
sgn(b*x + a) + 45*(x*e + d)^(7/2)*a*b^4*e^49*sgn(b*x + a) - 252*(x*e + d)^(5/2)*a*b^4*d*e^49*sgn(b*x + a) + 63
0*(x*e + d)^(3/2)*a*b^4*d^2*e^49*sgn(b*x + a) - 1260*sqrt(x*e + d)*a*b^4*d^3*e^49*sgn(b*x + a) + 126*(x*e + d)
^(5/2)*a^2*b^3*e^50*sgn(b*x + a) - 630*(x*e + d)^(3/2)*a^2*b^3*d*e^50*sgn(b*x + a) + 1890*sqrt(x*e + d)*a^2*b^
3*d^2*e^50*sgn(b*x + a) + 210*(x*e + d)^(3/2)*a^3*b^2*e^51*sgn(b*x + a) - 1260*sqrt(x*e + d)*a^3*b^2*d*e^51*sg
n(b*x + a) + 315*sqrt(x*e + d)*a^4*b*e^52*sgn(b*x + a))*e^(-54) + 2*(b^5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*e*sgn(
b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*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)/sqrt(x*e + d)