3.2102 \(\int (a+b x) \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=264 \[ \frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^5 (a+b x)}+\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^5 (a+b x)} \]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)) - (8*b*(b*d - a*e)^3*(d + e*
x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) + (12*b^2*(b*d - a*e)^2*(d + e*x)^(7/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) - (8*b^3*(b*d - a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^
5*(a + b*x)) + (2*b^4*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x))

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Rubi [A]  time = 0.104945, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 43} \[ \frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^5 (a+b x)}+\frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)) - (8*b*(b*d - a*e)^3*(d + e*
x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) + (12*b^2*(b*d - a*e)^2*(d + e*x)^(7/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) - (8*b^3*(b*d - a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^
5*(a + b*x)) + (2*b^4*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 (a+b x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^3 \sqrt{d+e x} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 \sqrt{d+e x} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^4 \sqrt{d+e x}}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^{3/2}}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{5/2}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{7/2}}{e^4}+\frac{b^4 (d+e x)^{9/2}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac{8 b (b d-a e)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}+\frac{12 b^2 (b d-a e)^2 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac{8 b^3 (b d-a e) (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac{2 b^4 (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0937492, size = 172, normalized size = 0.65 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (198 a^2 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+924 a^3 b e^3 (3 e x-2 d)+1155 a^4 e^4+44 a b^3 e \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(1155*a^4*e^4 + 924*a^3*b*e^3*(-2*d + 3*e*x) + 198*a^2*b^2*e^2*(8*d^2 - 1
2*d*e*x + 15*e^2*x^2) + 44*a*b^3*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + b^4*(128*d^4 - 192*d^3
*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)))/(3465*e^5*(a + b*x))

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Maple [A]  time = 0.006, size = 202, normalized size = 0.8 \begin{align*}{\frac{630\,{x}^{4}{b}^{4}{e}^{4}+3080\,{x}^{3}a{b}^{3}{e}^{4}-560\,{x}^{3}{b}^{4}d{e}^{3}+5940\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-2640\,{x}^{2}a{b}^{3}d{e}^{3}+480\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+5544\,x{a}^{3}b{e}^{4}-4752\,x{a}^{2}{b}^{2}d{e}^{3}+2112\,xa{b}^{3}{d}^{2}{e}^{2}-384\,x{b}^{4}{d}^{3}e+2310\,{a}^{4}{e}^{4}-3696\,d{e}^{3}{a}^{3}b+3168\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-1408\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{3465\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)*(e*x+d)^(1/2),x)

[Out]

2/3465*(e*x+d)^(3/2)*(315*b^4*e^4*x^4+1540*a*b^3*e^4*x^3-280*b^4*d*e^3*x^3+2970*a^2*b^2*e^4*x^2-1320*a*b^3*d*e
^3*x^2+240*b^4*d^2*e^2*x^2+2772*a^3*b*e^4*x-2376*a^2*b^2*d*e^3*x+1056*a*b^3*d^2*e^2*x-192*b^4*d^3*e*x+1155*a^4
*e^4-1848*a^3*b*d*e^3+1584*a^2*b^2*d^2*e^2-704*a*b^3*d^3*e+128*b^4*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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Maxima [B]  time = 1.14277, size = 518, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \,{\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} +{\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt{e x + d} a}{315 \, e^{4}} + \frac{2 \,{\left (315 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 528 \, a b^{2} d^{4} e + 792 \, a^{2} b d^{3} e^{2} - 462 \, a^{3} d^{2} e^{3} + 35 \,{\left (b^{3} d e^{4} + 33 \, a b^{2} e^{5}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{2} e^{3} - 33 \, a b^{2} d e^{4} - 297 \, a^{2} b e^{5}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{3} e^{2} - 66 \, a b^{2} d^{2} e^{3} + 99 \, a^{2} b d e^{4} + 231 \, a^{3} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{4} e - 264 \, a b^{2} d^{3} e^{2} + 396 \, a^{2} b d^{2} e^{3} - 231 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d} b}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*b^3*e^4*x^4 - 16*b^3*d^4 + 72*a*b^2*d^3*e - 126*a^2*b*d^2*e^2 + 105*a^3*d*e^3 + 5*(b^3*d*e^3 + 27*a*
b^2*e^4)*x^3 - 3*(2*b^3*d^2*e^2 - 9*a*b^2*d*e^3 - 63*a^2*b*e^4)*x^2 + (8*b^3*d^3*e - 36*a*b^2*d^2*e^2 + 63*a^2
*b*d*e^3 + 105*a^3*e^4)*x)*sqrt(e*x + d)*a/e^4 + 2/3465*(315*b^3*e^5*x^5 + 128*b^3*d^5 - 528*a*b^2*d^4*e + 792
*a^2*b*d^3*e^2 - 462*a^3*d^2*e^3 + 35*(b^3*d*e^4 + 33*a*b^2*e^5)*x^4 - 5*(8*b^3*d^2*e^3 - 33*a*b^2*d*e^4 - 297
*a^2*b*e^5)*x^3 + 3*(16*b^3*d^3*e^2 - 66*a*b^2*d^2*e^3 + 99*a^2*b*d*e^4 + 231*a^3*e^5)*x^2 - (64*b^3*d^4*e - 2
64*a*b^2*d^3*e^2 + 396*a^2*b*d^2*e^3 - 231*a^3*d*e^4)*x)*sqrt(e*x + d)*b/e^5

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Fricas [A]  time = 1.03265, size = 547, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (315 \, b^{4} e^{5} x^{5} + 128 \, b^{4} d^{5} - 704 \, a b^{3} d^{4} e + 1584 \, a^{2} b^{2} d^{3} e^{2} - 1848 \, a^{3} b d^{2} e^{3} + 1155 \, a^{4} d e^{4} + 35 \,{\left (b^{4} d e^{4} + 44 \, a b^{3} e^{5}\right )} x^{4} - 10 \,{\left (4 \, b^{4} d^{2} e^{3} - 22 \, a b^{3} d e^{4} - 297 \, a^{2} b^{2} e^{5}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{3} e^{2} - 44 \, a b^{3} d^{2} e^{3} + 99 \, a^{2} b^{2} d e^{4} + 462 \, a^{3} b e^{5}\right )} x^{2} -{\left (64 \, b^{4} d^{4} e - 352 \, a b^{3} d^{3} e^{2} + 792 \, a^{2} b^{2} d^{2} e^{3} - 924 \, a^{3} b d e^{4} - 1155 \, a^{4} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*b^4*e^5*x^5 + 128*b^4*d^5 - 704*a*b^3*d^4*e + 1584*a^2*b^2*d^3*e^2 - 1848*a^3*b*d^2*e^3 + 1155*a^4
*d*e^4 + 35*(b^4*d*e^4 + 44*a*b^3*e^5)*x^4 - 10*(4*b^4*d^2*e^3 - 22*a*b^3*d*e^4 - 297*a^2*b^2*e^5)*x^3 + 6*(8*
b^4*d^3*e^2 - 44*a*b^3*d^2*e^3 + 99*a^2*b^2*d*e^4 + 462*a^3*b*e^5)*x^2 - (64*b^4*d^4*e - 352*a*b^3*d^3*e^2 + 7
92*a^2*b^2*d^2*e^3 - 924*a^3*b*d*e^4 - 1155*a^4*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x\right ) \sqrt{d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)*(e*x+d)**(1/2),x)

[Out]

Integral((a + b*x)*sqrt(d + e*x)*((a + b*x)**2)**(3/2), x)

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Giac [A]  time = 1.18279, size = 332, normalized size = 1.26 \begin{align*} \frac{2}{3465} \,{\left (924 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{3} b e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 198 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a^{2} b^{2} e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + 44 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} a b^{3} e^{\left (-3\right )} \mathrm{sgn}\left (b x + a\right ) +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} b^{4} e^{\left (-4\right )} \mathrm{sgn}\left (b x + a\right ) + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} \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*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3465*(924*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*b*e^(-1)*sgn(b*x + a) + 198*(15*(x*e + d)^(7/2) - 42
*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b^2*e^(-2)*sgn(b*x + a) + 44*(35*(x*e + d)^(9/2) - 135*(x*e +
d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*b^3*e^(-3)*sgn(b*x + a) + (315*(x*e + d)^(1
1/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4
)*b^4*e^(-4)*sgn(b*x + a) + 1155*(x*e + d)^(3/2)*a^4*sgn(b*x + a))*e^(-1)