### 3.1570 $$\int (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx$$

Optimal. Leaf size=219 $\frac{4 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{9 b^5}+\frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{4 b^5}+\frac{4 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^4}{6 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}$

[Out]

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

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Rubi [A]  time = 0.0896996, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.036, Rules used = {645} $\frac{4 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{9 b^5}+\frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{4 b^5}+\frac{4 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^4}{6 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

Rule 645

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[ExpandLinearProduct[(b/2 + c*x)^(2*p), (d + e*x)^m, b
/2, c, x], 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] && IGtQ[m, 0] && EqQ[m - 2*p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.107724, size = 319, normalized size = 1.46 $\frac{x \sqrt{(a+b x)^2} \left (120 a^3 b^2 x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+45 a^2 b^3 x^3 \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )+210 a^4 b x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+252 a^5 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+10 a b^4 x^4 \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )+b^5 x^5 \left (945 d^2 e^2 x^2+720 d^3 e x+210 d^4+560 d e^3 x^3+126 e^4 x^4\right )\right )}{1260 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(252*a^5*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 210*a^4*b*x*(15*
d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 120*a^3*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^
2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 45*a^2*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^
3 + 35*e^4*x^4) + 10*a*b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + b^5*x^
5*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4)))/(1260*(a + b*x))

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Maple [B]  time = 0.157, size = 414, normalized size = 1.9 \begin{align*}{\frac{x \left ( 126\,{e}^{4}{b}^{5}{x}^{9}+700\,{x}^{8}{e}^{4}a{b}^{4}+560\,{x}^{8}d{e}^{3}{b}^{5}+1575\,{x}^{7}{e}^{4}{a}^{2}{b}^{3}+3150\,{x}^{7}d{e}^{3}a{b}^{4}+945\,{x}^{7}{d}^{2}{e}^{2}{b}^{5}+1800\,{x}^{6}{e}^{4}{a}^{3}{b}^{2}+7200\,{x}^{6}d{e}^{3}{a}^{2}{b}^{3}+5400\,{x}^{6}{d}^{2}{e}^{2}a{b}^{4}+720\,{x}^{6}{d}^{3}e{b}^{5}+1050\,{x}^{5}{e}^{4}{a}^{4}b+8400\,{x}^{5}d{e}^{3}{a}^{3}{b}^{2}+12600\,{x}^{5}{d}^{2}{e}^{2}{a}^{2}{b}^{3}+4200\,{x}^{5}{d}^{3}ea{b}^{4}+210\,{x}^{5}{d}^{4}{b}^{5}+252\,{x}^{4}{e}^{4}{a}^{5}+5040\,{x}^{4}d{e}^{3}{a}^{4}b+15120\,{x}^{4}{d}^{2}{e}^{2}{a}^{3}{b}^{2}+10080\,{x}^{4}{d}^{3}e{a}^{2}{b}^{3}+1260\,{x}^{4}{d}^{4}a{b}^{4}+1260\,{x}^{3}d{e}^{3}{a}^{5}+9450\,{x}^{3}{d}^{2}{e}^{2}{a}^{4}b+12600\,{x}^{3}{d}^{3}e{a}^{3}{b}^{2}+3150\,{x}^{3}{d}^{4}{a}^{2}{b}^{3}+2520\,{x}^{2}{d}^{2}{e}^{2}{a}^{5}+8400\,{x}^{2}{d}^{3}e{a}^{4}b+4200\,{x}^{2}{d}^{4}{a}^{3}{b}^{2}+2520\,x{d}^{3}e{a}^{5}+3150\,x{d}^{4}{a}^{4}b+1260\,{d}^{4}{a}^{5} \right ) }{1260\, \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)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/1260*x*(126*b^5*e^4*x^9+700*a*b^4*e^4*x^8+560*b^5*d*e^3*x^8+1575*a^2*b^3*e^4*x^7+3150*a*b^4*d*e^3*x^7+945*b^
5*d^2*e^2*x^7+1800*a^3*b^2*e^4*x^6+7200*a^2*b^3*d*e^3*x^6+5400*a*b^4*d^2*e^2*x^6+720*b^5*d^3*e*x^6+1050*a^4*b*
e^4*x^5+8400*a^3*b^2*d*e^3*x^5+12600*a^2*b^3*d^2*e^2*x^5+4200*a*b^4*d^3*e*x^5+210*b^5*d^4*x^5+252*a^5*e^4*x^4+
5040*a^4*b*d*e^3*x^4+15120*a^3*b^2*d^2*e^2*x^4+10080*a^2*b^3*d^3*e*x^4+1260*a*b^4*d^4*x^4+1260*a^5*d*e^3*x^3+9
450*a^4*b*d^2*e^2*x^3+12600*a^3*b^2*d^3*e*x^3+3150*a^2*b^3*d^4*x^3+2520*a^5*d^2*e^2*x^2+8400*a^4*b*d^3*e*x^2+4
200*a^3*b^2*d^4*x^2+2520*a^5*d^3*e*x+3150*a^4*b*d^4*x+1260*a^5*d^4)*((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)^4*(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 = 1.60779, size = 760, normalized size = 3.47 \begin{align*} \frac{1}{10} \, b^{5} e^{4} x^{10} + a^{5} d^{4} x + \frac{1}{9} \,{\left (4 \, b^{5} d e^{3} + 5 \, a b^{4} e^{4}\right )} x^{9} + \frac{1}{4} \,{\left (3 \, b^{5} d^{2} e^{2} + 10 \, a b^{4} d e^{3} + 5 \, a^{2} b^{3} e^{4}\right )} x^{8} + \frac{2}{7} \,{\left (2 \, b^{5} d^{3} e + 15 \, a b^{4} d^{2} e^{2} + 20 \, a^{2} b^{3} d e^{3} + 5 \, a^{3} b^{2} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{4} + 20 \, a b^{4} d^{3} e + 60 \, a^{2} b^{3} d^{2} e^{2} + 40 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (5 \, a b^{4} d^{4} + 40 \, a^{2} b^{3} d^{3} e + 60 \, a^{3} b^{2} d^{2} e^{2} + 20 \, a^{4} b d e^{3} + a^{5} e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (5 \, a^{2} b^{3} d^{4} + 20 \, a^{3} b^{2} d^{3} e + 15 \, a^{4} b d^{2} e^{2} + 2 \, a^{5} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (5 \, a^{3} b^{2} d^{4} + 10 \, a^{4} b d^{3} e + 3 \, a^{5} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{4} + 4 \, a^{5} d^{3} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{4} \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)**4*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((d + e*x)**4*((a + b*x)**2)**(5/2), x)

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Giac [B]  time = 1.24433, size = 761, normalized size = 3.47 \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)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

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