3.1970 \(\int (a+b x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=254 \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)}{9 e^5 (a+b x)}+\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^2}{4 e^5 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^3}{7 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^4}{6 e^5 (a+b x)} \]

[Out]

((b*d - a*e)^4*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^5*(a + b*x)) - (4*b*(b*d - a*e)^3*(d + e*x)^7*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) + (3*b^2*(b*d - a*e)^2*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(4*e^5*(a + b*x)) - (4*b^3*(b*d - a*e)*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (b^4
*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^5*(a + b*x))

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Rubi [A]  time = 0.270866, antiderivative size = 254, 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} \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)}{9 e^5 (a+b x)}+\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^2}{4 e^5 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^3}{7 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^4}{6 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.104784, size = 322, normalized size = 1.27 \[ \frac{x \sqrt{(a+b x)^2} \left (45 a^2 b^2 x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+120 a^3 b x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+210 a^4 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+10 a b^3 x^3 \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )+b^4 x^4 \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )\right )}{1260 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(210*a^4*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) +
120*a^3*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 45*a^2*b^2*
x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 10*a*b^3*x^3*(12
6*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + b^4*x^4*(252*d^5 + 105
0*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5)))/(1260*(a + b*x))

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Maple [B]  time = 0.008, size = 414, normalized size = 1.6 \begin{align*}{\frac{x \left ( 126\,{b}^{4}{e}^{5}{x}^{9}+560\,{x}^{8}a{b}^{3}{e}^{5}+700\,{x}^{8}{b}^{4}d{e}^{4}+945\,{x}^{7}{a}^{2}{b}^{2}{e}^{5}+3150\,{x}^{7}a{b}^{3}d{e}^{4}+1575\,{x}^{7}{b}^{4}{d}^{2}{e}^{3}+720\,{x}^{6}{a}^{3}b{e}^{5}+5400\,{x}^{6}{a}^{2}{b}^{2}d{e}^{4}+7200\,{x}^{6}a{b}^{3}{d}^{2}{e}^{3}+1800\,{x}^{6}{b}^{4}{d}^{3}{e}^{2}+210\,{x}^{5}{a}^{4}{e}^{5}+4200\,{x}^{5}{a}^{3}bd{e}^{4}+12600\,{x}^{5}{a}^{2}{b}^{2}{d}^{2}{e}^{3}+8400\,{x}^{5}a{b}^{3}{d}^{3}{e}^{2}+1050\,{x}^{5}{b}^{4}{d}^{4}e+1260\,{x}^{4}{a}^{4}d{e}^{4}+10080\,{x}^{4}{a}^{3}b{d}^{2}{e}^{3}+15120\,{x}^{4}{a}^{2}{b}^{2}{d}^{3}{e}^{2}+5040\,{x}^{4}a{b}^{3}{d}^{4}e+252\,{x}^{4}{b}^{4}{d}^{5}+3150\,{x}^{3}{a}^{4}{d}^{2}{e}^{3}+12600\,{x}^{3}{a}^{3}b{d}^{3}{e}^{2}+9450\,{x}^{3}{a}^{2}{b}^{2}{d}^{4}e+1260\,{x}^{3}a{b}^{3}{d}^{5}+4200\,{x}^{2}{a}^{4}{d}^{3}{e}^{2}+8400\,{x}^{2}{a}^{3}b{d}^{4}e+2520\,{x}^{2}{a}^{2}{b}^{2}{d}^{5}+3150\,x{a}^{4}{d}^{4}e+2520\,x{a}^{3}b{d}^{5}+1260\,{a}^{4}{d}^{5} \right ) }{1260\, \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)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/1260*x*(126*b^4*e^5*x^9+560*a*b^3*e^5*x^8+700*b^4*d*e^4*x^8+945*a^2*b^2*e^5*x^7+3150*a*b^3*d*e^4*x^7+1575*b^
4*d^2*e^3*x^7+720*a^3*b*e^5*x^6+5400*a^2*b^2*d*e^4*x^6+7200*a*b^3*d^2*e^3*x^6+1800*b^4*d^3*e^2*x^6+210*a^4*e^5
*x^5+4200*a^3*b*d*e^4*x^5+12600*a^2*b^2*d^2*e^3*x^5+8400*a*b^3*d^3*e^2*x^5+1050*b^4*d^4*e*x^5+1260*a^4*d*e^4*x
^4+10080*a^3*b*d^2*e^3*x^4+15120*a^2*b^2*d^3*e^2*x^4+5040*a*b^3*d^4*e*x^4+252*b^4*d^5*x^4+3150*a^4*d^2*e^3*x^3
+12600*a^3*b*d^3*e^2*x^3+9450*a^2*b^2*d^4*e*x^3+1260*a*b^3*d^5*x^3+4200*a^4*d^3*e^2*x^2+8400*a^3*b*d^4*e*x^2+2
520*a^2*b^2*d^5*x^2+3150*a^4*d^4*e*x+2520*a^3*b*d^5*x+1260*a^4*d^5)*((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)^5*(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 = 1.50792, size = 760, normalized size = 2.99 \begin{align*} \frac{1}{10} \, b^{4} e^{5} x^{10} + a^{4} d^{5} x + \frac{1}{9} \,{\left (5 \, b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} x^{9} + \frac{1}{4} \,{\left (5 \, b^{4} d^{2} e^{3} + 10 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} x^{8} + \frac{2}{7} \,{\left (5 \, b^{4} d^{3} e^{2} + 20 \, a b^{3} d^{2} e^{3} + 15 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (5 \, b^{4} d^{4} e + 40 \, a b^{3} d^{3} e^{2} + 60 \, a^{2} b^{2} d^{2} e^{3} + 20 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{5} + 20 \, a b^{3} d^{4} e + 60 \, a^{2} b^{2} d^{3} e^{2} + 40 \, a^{3} b d^{2} e^{3} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} d^{5} + 15 \, a^{2} b^{2} d^{4} e + 20 \, a^{3} b d^{3} e^{2} + 5 \, a^{4} d^{2} e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} d^{5} + 10 \, a^{3} b d^{4} e + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{5} + 5 \, a^{4} d^{4} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^4*e^5*x^10 + a^4*d^5*x + 1/9*(5*b^4*d*e^4 + 4*a*b^3*e^5)*x^9 + 1/4*(5*b^4*d^2*e^3 + 10*a*b^3*d*e^4 + 3*
a^2*b^2*e^5)*x^8 + 2/7*(5*b^4*d^3*e^2 + 20*a*b^3*d^2*e^3 + 15*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^7 + 1/6*(5*b^4*d^
4*e + 40*a*b^3*d^3*e^2 + 60*a^2*b^2*d^2*e^3 + 20*a^3*b*d*e^4 + a^4*e^5)*x^6 + 1/5*(b^4*d^5 + 20*a*b^3*d^4*e +
60*a^2*b^2*d^3*e^2 + 40*a^3*b*d^2*e^3 + 5*a^4*d*e^4)*x^5 + 1/2*(2*a*b^3*d^5 + 15*a^2*b^2*d^4*e + 20*a^3*b*d^3*
e^2 + 5*a^4*d^2*e^3)*x^4 + 2/3*(3*a^2*b^2*d^5 + 10*a^3*b*d^4*e + 5*a^4*d^3*e^2)*x^3 + 1/2*(4*a^3*b*d^5 + 5*a^4
*d^4*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 )^{5} \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)**5*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

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

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Giac [B]  time = 1.1392, size = 757, normalized size = 2.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^4*x^10*e^5*sgn(b*x + a) + 5/9*b^4*d*x^9*e^4*sgn(b*x + a) + 5/4*b^4*d^2*x^8*e^3*sgn(b*x + a) + 10/7*b^4*
d^3*x^7*e^2*sgn(b*x + a) + 5/6*b^4*d^4*x^6*e*sgn(b*x + a) + 1/5*b^4*d^5*x^5*sgn(b*x + a) + 4/9*a*b^3*x^9*e^5*s
gn(b*x + a) + 5/2*a*b^3*d*x^8*e^4*sgn(b*x + a) + 40/7*a*b^3*d^2*x^7*e^3*sgn(b*x + a) + 20/3*a*b^3*d^3*x^6*e^2*
sgn(b*x + a) + 4*a*b^3*d^4*x^5*e*sgn(b*x + a) + a*b^3*d^5*x^4*sgn(b*x + a) + 3/4*a^2*b^2*x^8*e^5*sgn(b*x + a)
+ 30/7*a^2*b^2*d*x^7*e^4*sgn(b*x + a) + 10*a^2*b^2*d^2*x^6*e^3*sgn(b*x + a) + 12*a^2*b^2*d^3*x^5*e^2*sgn(b*x +
a) + 15/2*a^2*b^2*d^4*x^4*e*sgn(b*x + a) + 2*a^2*b^2*d^5*x^3*sgn(b*x + a) + 4/7*a^3*b*x^7*e^5*sgn(b*x + a) +
10/3*a^3*b*d*x^6*e^4*sgn(b*x + a) + 8*a^3*b*d^2*x^5*e^3*sgn(b*x + a) + 10*a^3*b*d^3*x^4*e^2*sgn(b*x + a) + 20/
3*a^3*b*d^4*x^3*e*sgn(b*x + a) + 2*a^3*b*d^5*x^2*sgn(b*x + a) + 1/6*a^4*x^6*e^5*sgn(b*x + a) + a^4*d*x^5*e^4*s
gn(b*x + a) + 5/2*a^4*d^2*x^4*e^3*sgn(b*x + a) + 10/3*a^4*d^3*x^3*e^2*sgn(b*x + a) + 5/2*a^4*d^4*x^2*e*sgn(b*x
+ a) + a^4*d^5*x*sgn(b*x + a)