3.1554 \(\int (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=200 \[ \frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^4 (a+b x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)}{8 e^4 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2}{7 e^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3}{6 e^4 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.197707, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^4 (a+b x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)}{8 e^4 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2}{7 e^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3}{6 e^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0930337, size = 259, normalized size = 1.3 \[ \frac{x \sqrt{(a+b x)^2} \left (36 a^2 b x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+84 a^3 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+9 a b^2 x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+b^3 x^3 \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )\right )}{504 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(84*a^3*(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) +
36*a^2*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) + 9*a*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) + b^3*x^3*(126*d^5 + 50
4*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)))/(504*(a + b*x))

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Maple [B]  time = 0.156, size = 322, normalized size = 1.6 \begin{align*}{\frac{x \left ( 56\,{b}^{3}{e}^{5}{x}^{8}+189\,{x}^{7}{b}^{2}a{e}^{5}+315\,{x}^{7}{b}^{3}d{e}^{4}+216\,{x}^{6}b{a}^{2}{e}^{5}+1080\,{x}^{6}{b}^{2}ad{e}^{4}+720\,{x}^{6}{b}^{3}{d}^{2}{e}^{3}+84\,{x}^{5}{a}^{3}{e}^{5}+1260\,{x}^{5}b{a}^{2}d{e}^{4}+2520\,{x}^{5}{b}^{2}a{d}^{2}{e}^{3}+840\,{x}^{5}{b}^{3}{d}^{3}{e}^{2}+504\,{a}^{3}d{e}^{4}{x}^{4}+3024\,{a}^{2}b{d}^{2}{e}^{3}{x}^{4}+3024\,a{b}^{2}{d}^{3}{e}^{2}{x}^{4}+504\,{b}^{3}{d}^{4}e{x}^{4}+1260\,{x}^{3}{a}^{3}{d}^{2}{e}^{3}+3780\,{x}^{3}b{a}^{2}{d}^{3}{e}^{2}+1890\,{x}^{3}{b}^{2}a{d}^{4}e+126\,{x}^{3}{b}^{3}{d}^{5}+1680\,{x}^{2}{a}^{3}{d}^{3}{e}^{2}+2520\,{x}^{2}b{a}^{2}{d}^{4}e+504\,{x}^{2}{b}^{2}a{d}^{5}+1260\,x{a}^{3}{d}^{4}e+756\,xb{a}^{2}{d}^{5}+504\,{a}^{3}{d}^{5} \right ) }{504\, \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((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/504*x*(56*b^3*e^5*x^8+189*a*b^2*e^5*x^7+315*b^3*d*e^4*x^7+216*a^2*b*e^5*x^6+1080*a*b^2*d*e^4*x^6+720*b^3*d^2
*e^3*x^6+84*a^3*e^5*x^5+1260*a^2*b*d*e^4*x^5+2520*a*b^2*d^2*e^3*x^5+840*b^3*d^3*e^2*x^5+504*a^3*d*e^4*x^4+3024
*a^2*b*d^2*e^3*x^4+3024*a*b^2*d^3*e^2*x^4+504*b^3*d^4*e*x^4+1260*a^3*d^2*e^3*x^3+3780*a^2*b*d^3*e^2*x^3+1890*a
*b^2*d^4*e*x^3+126*b^3*d^5*x^3+1680*a^3*d^3*e^2*x^2+2520*a^2*b*d^4*e*x^2+504*a*b^2*d^5*x^2+1260*a^3*d^4*e*x+75
6*a^2*b*d^5*x+504*a^3*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((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 = 1.51822, size = 585, normalized size = 2.92 \begin{align*} \frac{1}{9} \, b^{3} e^{5} x^{9} + a^{3} d^{5} x + \frac{1}{8} \,{\left (5 \, b^{3} d e^{4} + 3 \, a b^{2} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, b^{3} d^{2} e^{3} + 15 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, b^{3} d^{3} e^{2} + 30 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} x^{6} +{\left (b^{3} d^{4} e + 6 \, a b^{2} d^{3} e^{2} + 6 \, a^{2} b d^{2} e^{3} + a^{3} d e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{5} + 15 \, a b^{2} d^{4} e + 30 \, a^{2} b d^{3} e^{2} + 10 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{2} d^{5} + 15 \, a^{2} b d^{4} e + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{5} + 5 \, a^{3} d^{4} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^3*e^5*x^9 + a^3*d^5*x + 1/8*(5*b^3*d*e^4 + 3*a*b^2*e^5)*x^8 + 1/7*(10*b^3*d^2*e^3 + 15*a*b^2*d*e^4 + 3*a
^2*b*e^5)*x^7 + 1/6*(10*b^3*d^3*e^2 + 30*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 + a^3*e^5)*x^6 + (b^3*d^4*e + 6*a*b^2*
d^3*e^2 + 6*a^2*b*d^2*e^3 + a^3*d*e^4)*x^5 + 1/4*(b^3*d^5 + 15*a*b^2*d^4*e + 30*a^2*b*d^3*e^2 + 10*a^3*d^2*e^3
)*x^4 + 1/3*(3*a*b^2*d^5 + 15*a^2*b*d^4*e + 10*a^3*d^3*e^2)*x^3 + 1/2*(3*a^2*b*d^5 + 5*a^3*d^4*e)*x^2

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

[Out]

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

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Giac [B]  time = 1.19842, size = 587, normalized size = 2.94 \begin{align*} \frac{1}{9} \, b^{3} x^{9} e^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{8} \, b^{3} d x^{8} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, b^{3} d^{2} x^{7} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, b^{3} d^{3} x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + b^{3} d^{4} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, b^{3} d^{5} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{8} \, a b^{2} x^{8} e^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{7} \, a b^{2} d x^{7} e^{4} \mathrm{sgn}\left (b x + a\right ) + 5 \, a b^{2} d^{2} x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{2} d^{3} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{4} \, a b^{2} d^{4} x^{4} e \mathrm{sgn}\left (b x + a\right ) + a b^{2} d^{5} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{7} \, a^{2} b x^{7} e^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b d x^{6} e^{4} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b d^{2} x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{2} \, a^{2} b d^{3} x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{2} b d^{4} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b d^{5} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, a^{3} x^{6} e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{3} d x^{5} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{3} d^{2} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{3} d^{3} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{3} d^{4} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{3} d^{5} x \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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