### 3.1569 $$\int (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx$$

Optimal. Leaf size=266 $\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac{10 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6}$

[Out]

((b*d - a*e)^5*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(7*b^6) + (5*e^2*(b*d - a*e)^3*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^6) + (10*e^
3*(b*d - a*e)^2*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^6) + (e^4*(b*d - a*e)*(a + b*x)^9*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(2*b^6) + (e^5*(a + b*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*b^6)

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Rubi [A]  time = 0.302778, antiderivative size = 266, 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} $\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac{10 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b*d - a*e)^5*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(7*b^6) + (5*e^2*(b*d - a*e)^3*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^6) + (10*e^
3*(b*d - a*e)^2*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^6) + (e^4*(b*d - a*e)*(a + b*x)^9*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(2*b^6) + (e^5*(a + b*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*b^6)

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 )^{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)^5 \, 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)^5 \left (a b+b^2 x\right )^5}{b^5}+\frac{5 e (b d-a e)^4 \left (a b+b^2 x\right )^6}{b^6}+\frac{10 e^2 (b d-a e)^3 \left (a b+b^2 x\right )^7}{b^7}+\frac{10 e^3 (b d-a e)^2 \left (a b+b^2 x\right )^8}{b^8}+\frac{5 e^4 (b d-a e) \left (a b+b^2 x\right )^9}{b^9}+\frac{e^5 \left (a b+b^2 x\right )^{10}}{b^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(b d-a e)^5 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^6}+\frac{5 e (b d-a e)^4 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac{5 e^2 (b d-a e)^3 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^6}+\frac{10 e^3 (b d-a e)^2 (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac{e^4 (b d-a e) (a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^6}+\frac{e^5 (a+b x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{11 b^6}\\ \end{align*}

Mathematica [A]  time = 0.132222, size = 385, normalized size = 1.45 $\frac{x \sqrt{(a+b x)^2} \left (165 a^3 b^2 x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+55 a^2 b^3 x^3 \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )+330 a^4 b x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+462 a^5 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+11 a b^4 x^4 \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )+b^5 x^5 \left (3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1980 d^4 e x+462 d^5+1386 d e^4 x^4+252 e^5 x^5\right )\right )}{2772 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(462*a^5*(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) +
330*a^4*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) + 165*a^3*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) + 55*a^2*b^3*x^3*
(126*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) + 11*a*b^4*x^4*(252*d
^5 + 1050*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) + b^5*x^5*(462*d^5 + 19
80*d^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5)))/(2772*(a + b*x))

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Maple [B]  time = 0.157, size = 506, normalized size = 1.9 \begin{align*}{\frac{x \left ( 252\,{b}^{5}{e}^{5}{x}^{10}+1386\,{x}^{9}a{b}^{4}{e}^{5}+1386\,{x}^{9}{b}^{5}d{e}^{4}+3080\,{x}^{8}{a}^{2}{b}^{3}{e}^{5}+7700\,{x}^{8}a{b}^{4}d{e}^{4}+3080\,{x}^{8}{b}^{5}{d}^{2}{e}^{3}+3465\,{x}^{7}{a}^{3}{b}^{2}{e}^{5}+17325\,{x}^{7}{a}^{2}{b}^{3}d{e}^{4}+17325\,{x}^{7}a{b}^{4}{d}^{2}{e}^{3}+3465\,{x}^{7}{b}^{5}{d}^{3}{e}^{2}+1980\,{x}^{6}{a}^{4}b{e}^{5}+19800\,{x}^{6}{a}^{3}{b}^{2}d{e}^{4}+39600\,{x}^{6}{a}^{2}{b}^{3}{d}^{2}{e}^{3}+19800\,{x}^{6}a{b}^{4}{d}^{3}{e}^{2}+1980\,{x}^{6}{b}^{5}{d}^{4}e+462\,{x}^{5}{a}^{5}{e}^{5}+11550\,{x}^{5}{a}^{4}bd{e}^{4}+46200\,{x}^{5}{a}^{3}{b}^{2}{d}^{2}{e}^{3}+46200\,{x}^{5}{a}^{2}{b}^{3}{d}^{3}{e}^{2}+11550\,{x}^{5}a{b}^{4}{d}^{4}e+462\,{x}^{5}{b}^{5}{d}^{5}+2772\,{a}^{5}d{e}^{4}{x}^{4}+27720\,{a}^{4}b{d}^{2}{e}^{3}{x}^{4}+55440\,{a}^{3}{b}^{2}{d}^{3}{e}^{2}{x}^{4}+27720\,{a}^{2}{b}^{3}{d}^{4}e{x}^{4}+2772\,a{b}^{4}{d}^{5}{x}^{4}+6930\,{x}^{3}{a}^{5}{d}^{2}{e}^{3}+34650\,{x}^{3}{a}^{4}b{d}^{3}{e}^{2}+34650\,{x}^{3}{a}^{3}{b}^{2}{d}^{4}e+6930\,{x}^{3}{a}^{2}{b}^{3}{d}^{5}+9240\,{x}^{2}{a}^{5}{d}^{3}{e}^{2}+23100\,{x}^{2}{a}^{4}b{d}^{4}e+9240\,{x}^{2}{a}^{3}{b}^{2}{d}^{5}+6930\,x{a}^{5}{d}^{4}e+6930\,x{a}^{4}b{d}^{5}+2772\,{a}^{5}{d}^{5} \right ) }{2772\, \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)^5*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/2772*x*(252*b^5*e^5*x^10+1386*a*b^4*e^5*x^9+1386*b^5*d*e^4*x^9+3080*a^2*b^3*e^5*x^8+7700*a*b^4*d*e^4*x^8+308
0*b^5*d^2*e^3*x^8+3465*a^3*b^2*e^5*x^7+17325*a^2*b^3*d*e^4*x^7+17325*a*b^4*d^2*e^3*x^7+3465*b^5*d^3*e^2*x^7+19
80*a^4*b*e^5*x^6+19800*a^3*b^2*d*e^4*x^6+39600*a^2*b^3*d^2*e^3*x^6+19800*a*b^4*d^3*e^2*x^6+1980*b^5*d^4*e*x^6+
462*a^5*e^5*x^5+11550*a^4*b*d*e^4*x^5+46200*a^3*b^2*d^2*e^3*x^5+46200*a^2*b^3*d^3*e^2*x^5+11550*a*b^4*d^4*e*x^
5+462*b^5*d^5*x^5+2772*a^5*d*e^4*x^4+27720*a^4*b*d^2*e^3*x^4+55440*a^3*b^2*d^3*e^2*x^4+27720*a^2*b^3*d^4*e*x^4
+2772*a*b^4*d^5*x^4+6930*a^5*d^2*e^3*x^3+34650*a^4*b*d^3*e^2*x^3+34650*a^3*b^2*d^4*e*x^3+6930*a^2*b^3*d^5*x^3+
9240*a^5*d^3*e^2*x^2+23100*a^4*b*d^4*e*x^2+9240*a^3*b^2*d^5*x^2+6930*a^5*d^4*e*x+6930*a^4*b*d^5*x+2772*a^5*d^5
)*((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)^5*(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 = 1.85984, size = 883, normalized size = 3.32 \begin{align*} \frac{1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac{1}{2} \,{\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac{5}{4} \,{\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac{5}{7} \,{\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} +{\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac{5}{2} \,{\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac{5}{2} \,{\left (a^{4} b d^{5} + a^{5} 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)^(5/2),x, algorithm="fricas")

[Out]

1/11*b^5*e^5*x^11 + a^5*d^5*x + 1/2*(b^5*d*e^4 + a*b^4*e^5)*x^10 + 5/9*(2*b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 2*a^2*
b^3*e^5)*x^9 + 5/4*(b^5*d^3*e^2 + 5*a*b^4*d^2*e^3 + 5*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^8 + 5/7*(b^5*d^4*e + 10*a
*b^4*d^3*e^2 + 20*a^2*b^3*d^2*e^3 + 10*a^3*b^2*d*e^4 + a^4*b*e^5)*x^7 + 1/6*(b^5*d^5 + 25*a*b^4*d^4*e + 100*a^
2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + a^5*e^5)*x^6 + (a*b^4*d^5 + 10*a^2*b^3*d^4*e + 20*a^3*b
^2*d^3*e^2 + 10*a^4*b*d^2*e^3 + a^5*d*e^4)*x^5 + 5/2*(a^2*b^3*d^5 + 5*a^3*b^2*d^4*e + 5*a^4*b*d^3*e^2 + a^5*d^
2*e^3)*x^4 + 5/3*(2*a^3*b^2*d^5 + 5*a^4*b*d^4*e + 2*a^5*d^3*e^2)*x^3 + 5/2*(a^4*b*d^5 + a^5*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{5}{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)**(5/2),x)

[Out]

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

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Giac [B]  time = 1.30252, size = 926, normalized size = 3.48 \begin{align*} \text{result too large to display} \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)^(5/2),x, algorithm="giac")

[Out]

1/11*b^5*x^11*e^5*sgn(b*x + a) + 1/2*b^5*d*x^10*e^4*sgn(b*x + a) + 10/9*b^5*d^2*x^9*e^3*sgn(b*x + a) + 5/4*b^5
*d^3*x^8*e^2*sgn(b*x + a) + 5/7*b^5*d^4*x^7*e*sgn(b*x + a) + 1/6*b^5*d^5*x^6*sgn(b*x + a) + 1/2*a*b^4*x^10*e^5
*sgn(b*x + a) + 25/9*a*b^4*d*x^9*e^4*sgn(b*x + a) + 25/4*a*b^4*d^2*x^8*e^3*sgn(b*x + a) + 50/7*a*b^4*d^3*x^7*e
^2*sgn(b*x + a) + 25/6*a*b^4*d^4*x^6*e*sgn(b*x + a) + a*b^4*d^5*x^5*sgn(b*x + a) + 10/9*a^2*b^3*x^9*e^5*sgn(b*
x + a) + 25/4*a^2*b^3*d*x^8*e^4*sgn(b*x + a) + 100/7*a^2*b^3*d^2*x^7*e^3*sgn(b*x + a) + 50/3*a^2*b^3*d^3*x^6*e
^2*sgn(b*x + a) + 10*a^2*b^3*d^4*x^5*e*sgn(b*x + a) + 5/2*a^2*b^3*d^5*x^4*sgn(b*x + a) + 5/4*a^3*b^2*x^8*e^5*s
gn(b*x + a) + 50/7*a^3*b^2*d*x^7*e^4*sgn(b*x + a) + 50/3*a^3*b^2*d^2*x^6*e^3*sgn(b*x + a) + 20*a^3*b^2*d^3*x^5
*e^2*sgn(b*x + a) + 25/2*a^3*b^2*d^4*x^4*e*sgn(b*x + a) + 10/3*a^3*b^2*d^5*x^3*sgn(b*x + a) + 5/7*a^4*b*x^7*e^
5*sgn(b*x + a) + 25/6*a^4*b*d*x^6*e^4*sgn(b*x + a) + 10*a^4*b*d^2*x^5*e^3*sgn(b*x + a) + 25/2*a^4*b*d^3*x^4*e^
2*sgn(b*x + a) + 25/3*a^4*b*d^4*x^3*e*sgn(b*x + a) + 5/2*a^4*b*d^5*x^2*sgn(b*x + a) + 1/6*a^5*x^6*e^5*sgn(b*x
+ a) + a^5*d*x^5*e^4*sgn(b*x + a) + 5/2*a^5*d^2*x^4*e^3*sgn(b*x + a) + 10/3*a^5*d^3*x^3*e^2*sgn(b*x + a) + 5/2
*a^5*d^4*x^2*e*sgn(b*x + a) + a^5*d^5*x*sgn(b*x + a)