\(\int x^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [164]

Optimal result

Integrand size = 24, antiderivative size = 181 \[ \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {a^4 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^5}-\frac {4 a^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {3 a^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^5}-\frac {4 a (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac {(a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^5} \]

[Out]

1/6*a^4*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^5-4/7*a^3*(b*x+a)^6*((b*x+a)^2)^(1/2)/b^5+3/4*a^2*(b*x+a)^7*((b*x+a)^2)^
(1/2)/b^5-4/9*a*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^5+1/10*(b*x+a)^9*((b*x+a)^2)^(1/2)/b^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {659} \[ \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac {4 a \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac {3 a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}+\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5}-\frac {4 a^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5} \]

[In]

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

[Out]

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

Rule 659

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

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.43 \[ \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x^5 \sqrt {(a+b x)^2} \left (252 a^5+1050 a^4 b x+1800 a^3 b^2 x^2+1575 a^2 b^3 x^3+700 a b^4 x^4+126 b^5 x^5\right )}{1260 (a+b x)} \]

[In]

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

[Out]

(x^5*Sqrt[(a + b*x)^2]*(252*a^5 + 1050*a^4*b*x + 1800*a^3*b^2*x^2 + 1575*a^2*b^3*x^3 + 700*a*b^4*x^4 + 126*b^5
*x^5))/(1260*(a + b*x))

Maple [A] (verified)

Time = 2.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.41

method	result	size
gosper	\(\frac {x^{5} \left (126 b^{5} x^{5}+700 a \,b^{4} x^{4}+1575 a^{2} b^{3} x^{3}+1800 a^{3} b^{2} x^{2}+1050 a^{4} b x +252 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{1260 \left (b x +a \right )^{5}}\)	\(74\)
default	\(\frac {x^{5} \left (126 b^{5} x^{5}+700 a \,b^{4} x^{4}+1575 a^{2} b^{3} x^{3}+1800 a^{3} b^{2} x^{2}+1050 a^{4} b x +252 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{1260 \left (b x +a \right )^{5}}\)	\(74\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{5} x^{5}}{5 b x +5 a}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a^{4} b \,x^{6}}{6 \left (b x +a \right )}+\frac {10 \sqrt {\left (b x +a \right )^{2}}\, a^{3} b^{2} x^{7}}{7 \left (b x +a \right )}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a^{2} b^{3} x^{8}}{4 \left (b x +a \right )}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a \,b^{4} x^{9}}{9 \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} x^{10}}{10 b x +10 a}\)	\(154\)

[In]

int(x^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/1260*x^5*(126*b^5*x^5+700*a*b^4*x^4+1575*a^2*b^3*x^3+1800*a^3*b^2*x^2+1050*a^4*b*x+252*a^5)*((b*x+a)^2)^(5/2
)/(b*x+a)^5

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.31 \[ \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{10} \, b^{5} x^{10} + \frac {5}{9} \, a b^{4} x^{9} + \frac {5}{4} \, a^{2} b^{3} x^{8} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{5} \, a^{5} x^{5} \]

[In]

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

[Out]

1/10*b^5*x^10 + 5/9*a*b^4*x^9 + 5/4*a^2*b^3*x^8 + 10/7*a^3*b^2*x^7 + 5/6*a^4*b*x^6 + 1/5*a^5*x^5

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.35 \[ \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{9}}{1260 b^{5}} - \frac {a^{8} x}{1260 b^{4}} + \frac {a^{7} x^{2}}{1260 b^{3}} - \frac {a^{6} x^{3}}{1260 b^{2}} + \frac {a^{5} x^{4}}{1260 b} + \frac {251 a^{4} x^{5}}{1260} + \frac {799 a^{3} b x^{6}}{1260} + \frac {143 a^{2} b^{2} x^{7}}{180} + \frac {41 a b^{3} x^{8}}{90} + \frac {b^{4} x^{9}}{10}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{8} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} - \frac {4 a^{6} \left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9} + \frac {6 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {11}{2}}}{11} - \frac {4 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {13}{2}}}{13} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {15}{2}}}{15}}{16 a^{5} b^{5}} & \text {for}\: a b \neq 0 \\\frac {x^{5} \left (a^{2}\right )^{\frac {5}{2}}}{5} & \text {otherwise} \end {cases} \]

[In]

integrate(x**4*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(a**9/(1260*b**5) - a**8*x/(1260*b**4) + a**7*x**2/(1260*b**3) - a
**6*x**3/(1260*b**2) + a**5*x**4/(1260*b) + 251*a**4*x**5/1260 + 799*a**3*b*x**6/1260 + 143*a**2*b**2*x**7/180
+ 41*a*b**3*x**8/90 + b**4*x**9/10), Ne(b**2, 0)), ((a**8*(a**2 + 2*a*b*x)**(7/2)/7 - 4*a**6*(a**2 + 2*a*b*x)
**(9/2)/9 + 6*a**4*(a**2 + 2*a*b*x)**(11/2)/11 - 4*a**2*(a**2 + 2*a*b*x)**(13/2)/13 + (a**2 + 2*a*b*x)**(15/2)
/15)/(16*a**5*b**5), Ne(a*b, 0)), (x**5*(a**2)**(5/2)/5, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.88 \[ \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{3}}{10 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} x}{6 \, b^{4}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x^{2}}{90 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5}}{6 \, b^{5}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} x}{180 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3}}{1260 \, b^{5}} \]

[In]

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

[Out]

1/10*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x/b^4 - 13/90*(b^2*x^2
+ 2*a*b*x + a^2)^(7/2)*a*x^2/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5/b^5 + 29/180*(b^2*x^2 + 2*a*b*x + a
^2)^(7/2)*a^2*x/b^4 - 209/1260*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3/b^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.59 \[ \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{10} \, b^{5} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{9} \, a b^{4} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{2} b^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, a^{3} b^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, a^{4} b x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{10} \mathrm {sgn}\left (b x + a\right )}{1260 \, b^{5}} \]

[In]

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

[Out]

1/10*b^5*x^10*sgn(b*x + a) + 5/9*a*b^4*x^9*sgn(b*x + a) + 5/4*a^2*b^3*x^8*sgn(b*x + a) + 10/7*a^3*b^2*x^7*sgn(
b*x + a) + 5/6*a^4*b*x^6*sgn(b*x + a) + 1/5*a^5*x^5*sgn(b*x + a) + 1/1260*a^10*sgn(b*x + a)/b^5

Mupad [F(-1)]

Timed out. \[ \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int x^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]

[In]

int(x^4*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int(x^4*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)