3.1994 \(\int (a+b x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=172 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{3 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{8 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^4} \]

[Out]

((b*d - a*e)^3*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^4) + (3*e*(b*d - a*e)^2*(a + b*x)^7*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(8*b^4) + (e^2*(b*d - a*e)*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^4) + (e^3*(a +
b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*b^4)

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Rubi [A]  time = 0.24426, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{3 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{8 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)^3*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^4) + (3*e*(b*d - a*e)^2*(a + b*x)^7*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(8*b^4) + (e^2*(b*d - a*e)*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^4) + (e^3*(a +
b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*b^4)

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

Mathematica [A]  time = 0.0995985, size = 294, normalized size = 1.71 \[ \frac{x \sqrt{(a+b x)^2} \left (210 a^4 b^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+120 a^3 b^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+45 a^2 b^4 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+252 a^5 b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+210 a^6 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+10 a b^5 x^5 \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )+b^6 x^6 \left (315 d^2 e x+120 d^3+280 d e^2 x^2+84 e^3 x^3\right )\right )}{840 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(210*a^6*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 252*a^5*b*x*(10*d^3 + 20*d^2*e*x +
15*d*e^2*x^2 + 4*e^3*x^3) + 210*a^4*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 120*a^3*b^3*x
^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 45*a^2*b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 +
35*e^3*x^3) + 10*a*b^5*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + b^6*x^6*(120*d^3 + 315*d^2*e*
x + 280*d*e^2*x^2 + 84*e^3*x^3)))/(840*(a + b*x))

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Maple [B]  time = 0.006, size = 380, normalized size = 2.2 \begin{align*}{\frac{x \left ( 84\,{e}^{3}{b}^{6}{x}^{9}+560\,{x}^{8}{e}^{3}{b}^{5}a+280\,{x}^{8}d{e}^{2}{b}^{6}+1575\,{x}^{7}{e}^{3}{a}^{2}{b}^{4}+1890\,{x}^{7}d{e}^{2}{b}^{5}a+315\,{x}^{7}{d}^{2}e{b}^{6}+2400\,{x}^{6}{e}^{3}{a}^{3}{b}^{3}+5400\,{x}^{6}d{e}^{2}{a}^{2}{b}^{4}+2160\,{x}^{6}{d}^{2}e{b}^{5}a+120\,{x}^{6}{d}^{3}{b}^{6}+2100\,{x}^{5}{e}^{3}{a}^{4}{b}^{2}+8400\,{x}^{5}d{e}^{2}{a}^{3}{b}^{3}+6300\,{x}^{5}{d}^{2}e{a}^{2}{b}^{4}+840\,{x}^{5}{d}^{3}{b}^{5}a+1008\,{x}^{4}{e}^{3}{a}^{5}b+7560\,{x}^{4}d{e}^{2}{a}^{4}{b}^{2}+10080\,{x}^{4}{d}^{2}e{a}^{3}{b}^{3}+2520\,{x}^{4}{d}^{3}{a}^{2}{b}^{4}+210\,{x}^{3}{e}^{3}{a}^{6}+3780\,{x}^{3}d{e}^{2}{a}^{5}b+9450\,{x}^{3}{d}^{2}e{a}^{4}{b}^{2}+4200\,{x}^{3}{d}^{3}{a}^{3}{b}^{3}+840\,{a}^{6}d{e}^{2}{x}^{2}+5040\,{a}^{5}b{d}^{2}e{x}^{2}+4200\,{a}^{4}{b}^{2}{d}^{3}{x}^{2}+1260\,x{d}^{2}e{a}^{6}+2520\,x{d}^{3}{a}^{5}b+840\,{d}^{3}{a}^{6} \right ) }{840\, \left ( bx+a \right ) ^{5}} \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*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/840*x*(84*b^6*e^3*x^9+560*a*b^5*e^3*x^8+280*b^6*d*e^2*x^8+1575*a^2*b^4*e^3*x^7+1890*a*b^5*d*e^2*x^7+315*b^6*
d^2*e*x^7+2400*a^3*b^3*e^3*x^6+5400*a^2*b^4*d*e^2*x^6+2160*a*b^5*d^2*e*x^6+120*b^6*d^3*x^6+2100*a^4*b^2*e^3*x^
5+8400*a^3*b^3*d*e^2*x^5+6300*a^2*b^4*d^2*e*x^5+840*a*b^5*d^3*x^5+1008*a^5*b*e^3*x^4+7560*a^4*b^2*d*e^2*x^4+10
080*a^3*b^3*d^2*e*x^4+2520*a^2*b^4*d^3*x^4+210*a^6*e^3*x^3+3780*a^5*b*d*e^2*x^3+9450*a^4*b^2*d^2*e*x^3+4200*a^
3*b^3*d^3*x^3+840*a^6*d*e^2*x^2+5040*a^5*b*d^2*e*x^2+4200*a^4*b^2*d^3*x^2+1260*a^6*d^2*e*x+2520*a^5*b*d^3*x+84
0*a^6*d^3)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.46748, size = 687, normalized size = 3.99 \begin{align*} \frac{1}{10} \, b^{6} e^{3} x^{10} + a^{6} d^{3} x + \frac{1}{3} \,{\left (b^{6} d e^{2} + 2 \, a b^{5} e^{3}\right )} x^{9} + \frac{3}{8} \,{\left (b^{6} d^{2} e + 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{3} + 18 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} + 20 \, a^{3} b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d^{3} + 15 \, a^{2} b^{4} d^{2} e + 20 \, a^{3} b^{3} d e^{2} + 5 \, a^{4} b^{2} e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (5 \, a^{2} b^{4} d^{3} + 20 \, a^{3} b^{3} d^{2} e + 15 \, a^{4} b^{2} d e^{2} + 2 \, a^{5} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} d^{3} + 45 \, a^{4} b^{2} d^{2} e + 18 \, a^{5} b d e^{2} + a^{6} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{3} + 6 \, a^{5} b d^{2} e + a^{6} d e^{2}\right )} x^{3} + \frac{3}{2} \,{\left (2 \, a^{5} b d^{3} + a^{6} d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/10*b^6*e^3*x^10 + a^6*d^3*x + 1/3*(b^6*d*e^2 + 2*a*b^5*e^3)*x^9 + 3/8*(b^6*d^2*e + 6*a*b^5*d*e^2 + 5*a^2*b^4
*e^3)*x^8 + 1/7*(b^6*d^3 + 18*a*b^5*d^2*e + 45*a^2*b^4*d*e^2 + 20*a^3*b^3*e^3)*x^7 + 1/2*(2*a*b^5*d^3 + 15*a^2
*b^4*d^2*e + 20*a^3*b^3*d*e^2 + 5*a^4*b^2*e^3)*x^6 + 3/5*(5*a^2*b^4*d^3 + 20*a^3*b^3*d^2*e + 15*a^4*b^2*d*e^2
+ 2*a^5*b*e^3)*x^5 + 1/4*(20*a^3*b^3*d^3 + 45*a^4*b^2*d^2*e + 18*a^5*b*d*e^2 + a^6*e^3)*x^4 + (5*a^4*b^2*d^3 +
6*a^5*b*d^2*e + a^6*d*e^2)*x^3 + 3/2*(2*a^5*b*d^3 + a^6*d^2*e)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x\right ) \left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**3*((a + b*x)**2)**(5/2), x)

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Giac [B]  time = 1.12151, size = 706, normalized size = 4.1 \begin{align*} \frac{1}{10} \, b^{6} x^{10} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, b^{6} d x^{9} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{8} \, b^{6} d^{2} x^{8} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{7} \, b^{6} d^{3} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, a b^{5} x^{9} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{9}{4} \, a b^{5} d x^{8} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{18}{7} \, a b^{5} d^{2} x^{7} e \mathrm{sgn}\left (b x + a\right ) + a b^{5} d^{3} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{8} \, a^{2} b^{4} x^{8} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{45}{7} \, a^{2} b^{4} d x^{7} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{2} \, a^{2} b^{4} d^{2} x^{6} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{7} \, a^{3} b^{3} x^{7} e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{3} b^{3} d x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + 12 \, a^{3} b^{3} d^{2} x^{5} e \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b^{2} x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + 9 \, a^{4} b^{2} d x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{45}{4} \, a^{4} b^{2} d^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{5} \, a^{5} b x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{9}{2} \, a^{5} b d x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{5} b d^{2} x^{3} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a^{6} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{6} d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{6} d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{6} d^{3} x \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/10*b^6*x^10*e^3*sgn(b*x + a) + 1/3*b^6*d*x^9*e^2*sgn(b*x + a) + 3/8*b^6*d^2*x^8*e*sgn(b*x + a) + 1/7*b^6*d^3
*x^7*sgn(b*x + a) + 2/3*a*b^5*x^9*e^3*sgn(b*x + a) + 9/4*a*b^5*d*x^8*e^2*sgn(b*x + a) + 18/7*a*b^5*d^2*x^7*e*s
gn(b*x + a) + a*b^5*d^3*x^6*sgn(b*x + a) + 15/8*a^2*b^4*x^8*e^3*sgn(b*x + a) + 45/7*a^2*b^4*d*x^7*e^2*sgn(b*x
+ a) + 15/2*a^2*b^4*d^2*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^3*x^5*sgn(b*x + a) + 20/7*a^3*b^3*x^7*e^3*sgn(b*x + a
) + 10*a^3*b^3*d*x^6*e^2*sgn(b*x + a) + 12*a^3*b^3*d^2*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^3*x^4*sgn(b*x + a) + 5
/2*a^4*b^2*x^6*e^3*sgn(b*x + a) + 9*a^4*b^2*d*x^5*e^2*sgn(b*x + a) + 45/4*a^4*b^2*d^2*x^4*e*sgn(b*x + a) + 5*a
^4*b^2*d^3*x^3*sgn(b*x + a) + 6/5*a^5*b*x^5*e^3*sgn(b*x + a) + 9/2*a^5*b*d*x^4*e^2*sgn(b*x + a) + 6*a^5*b*d^2*
x^3*e*sgn(b*x + a) + 3*a^5*b*d^3*x^2*sgn(b*x + a) + 1/4*a^6*x^4*e^3*sgn(b*x + a) + a^6*d*x^3*e^2*sgn(b*x + a)
+ 3/2*a^6*d^2*x^2*e*sgn(b*x + a) + a^6*d^3*x*sgn(b*x + a)