3.1695 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=316 \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^6 (a+b x)}-\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}{e^6 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) + (10*b*(b*d - a*e)^4*(d + e*x)
^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)) - (4*b^2*(b*d - a*e)^3*(d + e*x)^(5/2)*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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*
(a + b*x)) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)) + (2*b^5*(d
+ e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.0966293, 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} \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^6 (a+b x)}-\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}{e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) + (10*b*(b*d - a*e)^4*(d + e*x)
^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)) - (4*b^2*(b*d - a*e)^3*(d + e*x)^(5/2)*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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*
(a + b*x)) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)) + (2*b^5*(d
+ e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*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}}{\sqrt{d+e x}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{\sqrt{d+e x}} \, 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 \sqrt{d+e x}}+\frac{5 b^6 (b d-a e)^4 \sqrt{d+e x}}{e^5}-\frac{10 b^7 (b d-a e)^3 (d+e x)^{3/2}}{e^5}+\frac{10 b^8 (b d-a e)^2 (d+e x)^{5/2}}{e^5}-\frac{5 b^9 (b d-a e) (d+e x)^{7/2}}{e^5}+\frac{b^{10} (d+e x)^{9/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{2 (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac{10 b (b d-a e)^4 (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^2 (b d-a e)^3 (d+e x)^{5/2} \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)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}-\frac{10 b^4 (b d-a e) (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}+\frac{2 b^5 (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.12402, size = 234, normalized size = 0.74 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (198 a^2 b^3 e^2 \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )+462 a^3 b^2 e^3 \left (8 d^2-4 d e x+3 e^2 x^2\right )+1155 a^4 b e^4 (e x-2 d)+693 a^5 e^5+11 a b^4 e \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (-96 d^3 e^2 x^2+80 d^2 e^3 x^3+128 d^4 e x-256 d^5-70 d e^4 x^4+63 e^5 x^5\right )\right )}{693 e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(693*a^5*e^5 + 1155*a^4*b*e^4*(-2*d + e*x) + 462*a^3*b^2*e^3*(8*d^2 - 4*d*e
*x + 3*e^2*x^2) + 198*a^2*b^3*e^2*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 11*a*b^4*e*(128*d^4 - 64*d
^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + b^5*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2 + 80*d^2*e
^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5)))/(693*e^6*(a + b*x))

________________________________________________________________________________________

Maple [A]  time = 0.155, size = 289, normalized size = 0.9 \begin{align*}{\frac{126\,{x}^{5}{b}^{5}{e}^{5}+770\,{x}^{4}a{b}^{4}{e}^{5}-140\,{x}^{4}{b}^{5}d{e}^{4}+1980\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-880\,{x}^{3}a{b}^{4}d{e}^{4}+160\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+2772\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-2376\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+1056\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-192\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+2310\,x{a}^{4}b{e}^{5}-3696\,x{a}^{3}{b}^{2}d{e}^{4}+3168\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-1408\,xa{b}^{4}{d}^{3}{e}^{2}+256\,x{b}^{5}{d}^{4}e+1386\,{a}^{5}{e}^{5}-4620\,d{e}^{4}{a}^{4}b+7392\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-6336\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+2816\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{693\, \left ( bx+a \right ) ^{5}{e}^{6}}\sqrt{ex+d} \left ( \left ( bx+a \right ) ^{2} \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)^(1/2),x)

[Out]

2/693*(e*x+d)^(1/2)*(63*b^5*e^5*x^5+385*a*b^4*e^5*x^4-70*b^5*d*e^4*x^4+990*a^2*b^3*e^5*x^3-440*a*b^4*d*e^4*x^3
+80*b^5*d^2*e^3*x^3+1386*a^3*b^2*e^5*x^2-1188*a^2*b^3*d*e^4*x^2+528*a*b^4*d^2*e^3*x^2-96*b^5*d^3*e^2*x^2+1155*
a^4*b*e^5*x-1848*a^3*b^2*d*e^4*x+1584*a^2*b^3*d^2*e^3*x-704*a*b^4*d^3*e^2*x+128*b^5*d^4*e*x+693*a^5*e^5-2310*a
^4*b*d*e^4+3696*a^3*b^2*d^2*e^3-3168*a^2*b^3*d^3*e^2+1408*a*b^4*d^4*e-256*b^5*d^5)*((b*x+a)^2)^(5/2)/e^6/(b*x+
a)^5

________________________________________________________________________________________

Maxima [A]  time = 1.09507, size = 456, normalized size = 1.44 \begin{align*} \frac{2 \,{\left (63 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1408 \, a b^{4} d^{5} e - 3168 \, a^{2} b^{3} d^{4} e^{2} + 3696 \, a^{3} b^{2} d^{3} e^{3} - 2310 \, a^{4} b d^{2} e^{4} + 693 \, a^{5} d e^{5} - 7 \,{\left (b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 5 \,{\left (2 \, b^{5} d^{2} e^{4} - 11 \, a b^{4} d e^{5} + 198 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \,{\left (8 \, b^{5} d^{3} e^{3} - 44 \, a b^{4} d^{2} e^{4} + 99 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} +{\left (32 \, b^{5} d^{4} e^{2} - 176 \, a b^{4} d^{3} e^{3} + 396 \, a^{2} b^{3} d^{2} e^{4} - 462 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} -{\left (128 \, b^{5} d^{5} e - 704 \, a b^{4} d^{4} e^{2} + 1584 \, a^{2} b^{3} d^{3} e^{3} - 1848 \, a^{3} b^{2} d^{2} e^{4} + 1155 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )}}{693 \, \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)^(1/2),x, algorithm="maxima")

[Out]

2/693*(63*b^5*e^6*x^6 - 256*b^5*d^6 + 1408*a*b^4*d^5*e - 3168*a^2*b^3*d^4*e^2 + 3696*a^3*b^2*d^3*e^3 - 2310*a^
4*b*d^2*e^4 + 693*a^5*d*e^5 - 7*(b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 5*(2*b^5*d^2*e^4 - 11*a*b^4*d*e^5 + 198*a^2*b
^3*e^6)*x^4 - 2*(8*b^5*d^3*e^3 - 44*a*b^4*d^2*e^4 + 99*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (32*b^5*d^4*e^2
- 176*a*b^4*d^3*e^3 + 396*a^2*b^3*d^2*e^4 - 462*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (128*b^5*d^5*e - 704*a*b
^4*d^4*e^2 + 1584*a^2*b^3*d^3*e^3 - 1848*a^3*b^2*d^2*e^4 + 1155*a^4*b*d*e^5 - 693*a^5*e^6)*x)/(sqrt(e*x + d)*e
^6)

________________________________________________________________________________________

Fricas [A]  time = 1.60847, size = 586, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 1408 \, a b^{4} d^{4} e - 3168 \, a^{2} b^{3} d^{3} e^{2} + 3696 \, a^{3} b^{2} d^{2} e^{3} - 2310 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} - 35 \,{\left (2 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 44 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 88 \, a b^{4} d^{2} e^{3} + 198 \, a^{2} b^{3} d e^{4} - 231 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 704 \, a b^{4} d^{3} e^{2} + 1584 \, a^{2} b^{3} d^{2} e^{3} - 1848 \, a^{3} b^{2} d e^{4} + 1155 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{693 \, 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)^(1/2),x, algorithm="fricas")

[Out]

2/693*(63*b^5*e^5*x^5 - 256*b^5*d^5 + 1408*a*b^4*d^4*e - 3168*a^2*b^3*d^3*e^2 + 3696*a^3*b^2*d^2*e^3 - 2310*a^
4*b*d*e^4 + 693*a^5*e^5 - 35*(2*b^5*d*e^4 - 11*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 44*a*b^4*d*e^4 + 99*a^2*b^
3*e^5)*x^3 - 6*(16*b^5*d^3*e^2 - 88*a*b^4*d^2*e^3 + 198*a^2*b^3*d*e^4 - 231*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e
- 704*a*b^4*d^3*e^2 + 1584*a^2*b^3*d^2*e^3 - 1848*a^3*b^2*d*e^4 + 1155*a^4*b*e^5)*x)*sqrt(e*x + d)/e^6

________________________________________________________________________________________

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)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.18321, size = 451, normalized size = 1.43 \begin{align*} \frac{2}{693} \,{\left (1155 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{4} b e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a^{3} b^{2} e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + 198 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} a^{2} b^{3} e^{\left (-3\right )} \mathrm{sgn}\left (b x + a\right ) + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} a b^{4} e^{\left (-4\right )} \mathrm{sgn}\left (b x + a\right ) +{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{x e + d} d^{5}\right )} b^{5} e^{\left (-5\right )} \mathrm{sgn}\left (b x + a\right ) + 693 \, \sqrt{x e + d} a^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\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)^(1/2),x, algorithm="giac")

[Out]

2/693*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*b*e^(-1)*sgn(b*x + a) + 462*(3*(x*e + d)^(5/2) - 10*(x*e
+ d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b^2*e^(-2)*sgn(b*x + a) + 198*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2
)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*b^3*e^(-3)*sgn(b*x + a) + 11*(35*(x*e + d)^(9/2) - 18
0*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^4*e^(-4)*
sgn(b*x + a) + (63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d
^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^5*e^(-5)*sgn(b*x + a) + 693*sqrt(x*e + d)*a^5*sgn(b*x
+ a))*e^(-1)