### 3.1571 $$\int (d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx$$

Optimal. Leaf size=172 $\frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{8 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{6 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}$

[Out]

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

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Rubi [A]  time = 0.180173, antiderivative size = 172, 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} $\frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{8 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{6 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}$

Antiderivative was successfully veriﬁed.

[In]

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

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

Mathematica [A]  time = 0.0872429, size = 253, normalized size = 1.47 $\frac{x \sqrt{(a+b x)^2} \left (84 a^3 b^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+36 a^2 b^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+126 a^4 b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+126 a^5 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+9 a b^4 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+b^5 x^5 \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )\right )}{504 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(126*a^5*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 126*a^4*b*x*(10*d^3 + 20*d^2*e*x +
15*d*e^2*x^2 + 4*e^3*x^3) + 84*a^3*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 36*a^2*b^3*x^3
*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 9*a*b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^
3*x^3) + b^5*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)))/(504*(a + b*x))

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Maple [B]  time = 0.156, size = 322, normalized size = 1.9 \begin{align*}{\frac{x \left ( 56\,{e}^{3}{b}^{5}{x}^{8}+315\,{x}^{7}{e}^{3}a{b}^{4}+189\,{x}^{7}d{e}^{2}{b}^{5}+720\,{x}^{6}{e}^{3}{a}^{2}{b}^{3}+1080\,{x}^{6}d{e}^{2}a{b}^{4}+216\,{x}^{6}{d}^{2}e{b}^{5}+840\,{x}^{5}{e}^{3}{a}^{3}{b}^{2}+2520\,{x}^{5}d{e}^{2}{a}^{2}{b}^{3}+1260\,{x}^{5}{d}^{2}ea{b}^{4}+84\,{x}^{5}{d}^{3}{b}^{5}+504\,{a}^{4}b{e}^{3}{x}^{4}+3024\,{a}^{3}{b}^{2}d{e}^{2}{x}^{4}+3024\,{a}^{2}{b}^{3}{d}^{2}e{x}^{4}+504\,a{b}^{4}{d}^{3}{x}^{4}+126\,{x}^{3}{e}^{3}{a}^{5}+1890\,{x}^{3}d{e}^{2}{a}^{4}b+3780\,{x}^{3}{d}^{2}e{a}^{3}{b}^{2}+1260\,{x}^{3}{d}^{3}{a}^{2}{b}^{3}+504\,{x}^{2}d{e}^{2}{a}^{5}+2520\,{x}^{2}{d}^{2}e{a}^{4}b+1680\,{x}^{2}{d}^{3}{a}^{3}{b}^{2}+756\,x{d}^{2}e{a}^{5}+1260\,x{d}^{3}{a}^{4}b+504\,{d}^{3}{a}^{5} \right ) }{504\, \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((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/504*x*(56*b^5*e^3*x^8+315*a*b^4*e^3*x^7+189*b^5*d*e^2*x^7+720*a^2*b^3*e^3*x^6+1080*a*b^4*d*e^2*x^6+216*b^5*d
^2*e*x^6+840*a^3*b^2*e^3*x^5+2520*a^2*b^3*d*e^2*x^5+1260*a*b^4*d^2*e*x^5+84*b^5*d^3*x^5+504*a^4*b*e^3*x^4+3024
*a^3*b^2*d*e^2*x^4+3024*a^2*b^3*d^2*e*x^4+504*a*b^4*d^3*x^4+126*a^5*e^3*x^3+1890*a^4*b*d*e^2*x^3+3780*a^3*b^2*
d^2*e*x^3+1260*a^2*b^3*d^3*x^3+504*a^5*d*e^2*x^2+2520*a^4*b*d^2*e*x^2+1680*a^3*b^2*d^3*x^2+756*a^5*d^2*e*x+126
0*a^4*b*d^3*x+504*a^5*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((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 = 1.49215, size = 585, normalized size = 3.4 \begin{align*} \frac{1}{9} \, b^{5} e^{3} x^{9} + a^{5} d^{3} x + \frac{1}{8} \,{\left (3 \, b^{5} d e^{2} + 5 \, a b^{4} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, b^{5} d^{2} e + 15 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{3} + 15 \, a b^{4} d^{2} e + 30 \, a^{2} b^{3} d e^{2} + 10 \, a^{3} b^{2} e^{3}\right )} x^{6} +{\left (a b^{4} d^{3} + 6 \, a^{2} b^{3} d^{2} e + 6 \, a^{3} b^{2} d e^{2} + a^{4} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (10 \, a^{2} b^{3} d^{3} + 30 \, a^{3} b^{2} d^{2} e + 15 \, a^{4} b d e^{2} + a^{5} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} d^{3} + 15 \, a^{4} b d^{2} e + 3 \, a^{5} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{3} + 3 \, a^{5} d^{2} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [B]  time = 1.19104, size = 595, normalized size = 3.46 \begin{align*} \frac{1}{9} \, b^{5} x^{9} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{8} \, b^{5} d x^{8} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{7} \, b^{5} d^{2} x^{7} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, b^{5} d^{3} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{8} \, a b^{4} x^{8} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{7} \, a b^{4} d x^{7} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a b^{4} d^{2} x^{6} e \mathrm{sgn}\left (b x + a\right ) + a b^{4} d^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, a^{2} b^{3} x^{7} e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{2} b^{3} d x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{3} d^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a^{3} b^{2} x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{2} \, a^{3} b^{2} d^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{2} d^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + a^{4} b x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{4} \, a^{4} b d x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b d^{2} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b d^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a^{5} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{5} d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{5} d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{5} d^{3} 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)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

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