3.1972 \(\int (a+b x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=172 \[ \frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^4}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{2 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^3}{5 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4} \]

[Out]

((b*d - a*e)^3*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*b^4) + (e*(b*d - a*e)^2*(a + b*x)^5*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(2*b^4) + (3*e^2*(b*d - a*e)*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^4) + (e^3*(a +
b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*b^4)

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Rubi [A]  time = 0.179733, 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} \[ \frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^4}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{2 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^3}{5 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0731514, size = 212, normalized size = 1.23 \[ \frac{x \sqrt{(a+b x)^2} \left (28 a^2 b^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+56 a^3 b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+70 a^4 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+8 a b^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+b^4 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )\right )}{280 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(70*a^4*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 56*a^3*b*x*(10*d^3 + 20*d^2*e*x + 1
5*d*e^2*x^2 + 4*e^3*x^3) + 28*a^2*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 8*a*b^3*x^3*(35*
d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3)))/
(280*(a + b*x))

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Maple [B]  time = 0.007, size = 264, normalized size = 1.5 \begin{align*}{\frac{x \left ( 35\,{e}^{3}{b}^{4}{x}^{7}+160\,{x}^{6}{e}^{3}a{b}^{3}+120\,{x}^{6}d{e}^{2}{b}^{4}+280\,{x}^{5}{e}^{3}{a}^{2}{b}^{2}+560\,{x}^{5}d{e}^{2}a{b}^{3}+140\,{x}^{5}{d}^{2}e{b}^{4}+224\,{x}^{4}{e}^{3}{a}^{3}b+1008\,{x}^{4}d{e}^{2}{a}^{2}{b}^{2}+672\,{x}^{4}{d}^{2}ea{b}^{3}+56\,{x}^{4}{d}^{3}{b}^{4}+70\,{x}^{3}{e}^{3}{a}^{4}+840\,{x}^{3}d{e}^{2}{a}^{3}b+1260\,{x}^{3}{d}^{2}e{a}^{2}{b}^{2}+280\,{x}^{3}{d}^{3}a{b}^{3}+280\,{a}^{4}d{e}^{2}{x}^{2}+1120\,{a}^{3}b{d}^{2}e{x}^{2}+560\,{a}^{2}{b}^{2}{d}^{3}{x}^{2}+420\,x{d}^{2}e{a}^{4}+560\,b{d}^{3}{a}^{3}x+280\,{d}^{3}{a}^{4} \right ) }{280\, \left ( bx+a \right ) ^{3}} \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)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/280*x*(35*b^4*e^3*x^7+160*a*b^3*e^3*x^6+120*b^4*d*e^2*x^6+280*a^2*b^2*e^3*x^5+560*a*b^3*d*e^2*x^5+140*b^4*d^
2*e*x^5+224*a^3*b*e^3*x^4+1008*a^2*b^2*d*e^2*x^4+672*a*b^3*d^2*e*x^4+56*b^4*d^3*x^4+70*a^4*e^3*x^3+840*a^3*b*d
*e^2*x^3+1260*a^2*b^2*d^2*e*x^3+280*a*b^3*d^3*x^3+280*a^4*d*e^2*x^2+1120*a^3*b*d^2*e*x^2+560*a^2*b^2*d^3*x^2+4
20*a^4*d^2*e*x+560*a^3*b*d^3*x+280*a^4*d^3)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51019, size = 471, normalized size = 2.74 \begin{align*} \frac{1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac{1}{7} \,{\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{3} + 3 \, a^{4} 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)^(3/2),x, algorithm="fricas")

[Out]

1/8*b^4*e^3*x^8 + a^4*d^3*x + 1/7*(3*b^4*d*e^2 + 4*a*b^3*e^3)*x^7 + 1/2*(b^4*d^2*e + 4*a*b^3*d*e^2 + 2*a^2*b^2
*e^3)*x^6 + 1/5*(b^4*d^3 + 12*a*b^3*d^2*e + 18*a^2*b^2*d*e^2 + 4*a^3*b*e^3)*x^5 + 1/4*(4*a*b^3*d^3 + 18*a^2*b^
2*d^2*e + 12*a^3*b*d*e^2 + a^4*e^3)*x^4 + (2*a^2*b^2*d^3 + 4*a^3*b*d^2*e + a^4*d*e^2)*x^3 + 1/2*(4*a^3*b*d^3 +
3*a^4*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{3}{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)**(3/2),x)

[Out]

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

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Giac [B]  time = 1.2082, size = 486, normalized size = 2.83 \begin{align*} \frac{1}{8} \, b^{4} x^{8} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{7} \, b^{4} d x^{7} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, b^{4} d^{2} x^{6} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, b^{4} d^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{4}{7} \, a b^{3} x^{7} e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, a b^{3} d x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{12}{5} \, a b^{3} d^{2} x^{5} e \mathrm{sgn}\left (b x + a\right ) + a b^{3} d^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{2} x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{18}{5} \, a^{2} b^{2} d x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{9}{2} \, a^{2} b^{2} d^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{4}{5} \, a^{3} b x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{3} b d x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{3} b d^{2} x^{3} e \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a^{4} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{4} d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{4} d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{4} 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)^(3/2),x, algorithm="giac")

[Out]

1/8*b^4*x^8*e^3*sgn(b*x + a) + 3/7*b^4*d*x^7*e^2*sgn(b*x + a) + 1/2*b^4*d^2*x^6*e*sgn(b*x + a) + 1/5*b^4*d^3*x
^5*sgn(b*x + a) + 4/7*a*b^3*x^7*e^3*sgn(b*x + a) + 2*a*b^3*d*x^6*e^2*sgn(b*x + a) + 12/5*a*b^3*d^2*x^5*e*sgn(b
*x + a) + a*b^3*d^3*x^4*sgn(b*x + a) + a^2*b^2*x^6*e^3*sgn(b*x + a) + 18/5*a^2*b^2*d*x^5*e^2*sgn(b*x + a) + 9/
2*a^2*b^2*d^2*x^4*e*sgn(b*x + a) + 2*a^2*b^2*d^3*x^3*sgn(b*x + a) + 4/5*a^3*b*x^5*e^3*sgn(b*x + a) + 3*a^3*b*d
*x^4*e^2*sgn(b*x + a) + 4*a^3*b*d^2*x^3*e*sgn(b*x + a) + 2*a^3*b*d^3*x^2*sgn(b*x + a) + 1/4*a^4*x^4*e^3*sgn(b*
x + a) + a^4*d*x^3*e^2*sgn(b*x + a) + 3/2*a^4*d^2*x^2*e*sgn(b*x + a) + a^4*d^3*x*sgn(b*x + a)