\(\int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx\) [39]

Optimal result

Integrand size = 24, antiderivative size = 150 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\frac {21}{16} a^4 x \sqrt {a^2-b^2 x^2}-\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}-\frac {7 a^2 (8 a+3 b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}+\frac {21 a^6 \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{16 b} \] Output:

21/16*a^4*x*(-b^2*x^2+a^2)^(1/2)-3/10*a*(b*x+a)^2*(-b^2*x^2+a^2)^(3/2)/b-1 
/6*(b*x+a)^3*(-b^2*x^2+a^2)^(3/2)/b-7/40*a^2*(3*b*x+8*a)*(-b^2*x^2+a^2)^(3 
/2)/b+21/16*a^6*arctan(b*x/(-b^2*x^2+a^2)^(1/2))/b
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.76 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\frac {\sqrt {a^2-b^2 x^2} \left (-448 a^5-75 a^4 b x+256 a^3 b^2 x^2+350 a^2 b^3 x^3+192 a b^4 x^4+40 b^5 x^5\right )-630 a^6 \arctan \left (\frac {b x}{\sqrt {a^2}-\sqrt {a^2-b^2 x^2}}\right )}{240 b} \] Input:

Integrate[(a + b*x)^4*Sqrt[a^2 - b^2*x^2],x]
 

Output:

(Sqrt[a^2 - b^2*x^2]*(-448*a^5 - 75*a^4*b*x + 256*a^3*b^2*x^2 + 350*a^2*b^ 
3*x^3 + 192*a*b^4*x^4 + 40*b^5*x^5) - 630*a^6*ArcTan[(b*x)/(Sqrt[a^2] - Sq 
rt[a^2 - b^2*x^2])])/(240*b)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {469, 469, 469, 455, 211, 224, 216}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {3}{2} a \int (a+b x)^3 \sqrt {a^2-b^2 x^2}dx-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \int (a+b x)^2 \sqrt {a^2-b^2 x^2}dx-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{5 b}\right )-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \int (a+b x) \sqrt {a^2-b^2 x^2}dx-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}\right )-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{5 b}\right )-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \int \sqrt {a^2-b^2 x^2}dx-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}\right )-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{5 b}\right )-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {1}{2} a^2 \int \frac {1}{\sqrt {a^2-b^2 x^2}}dx+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}\right )-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{5 b}\right )-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {1}{2} a^2 \int \frac {1}{\frac {b^2 x^2}{a^2-b^2 x^2}+1}d\frac {x}{\sqrt {a^2-b^2 x^2}}+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}\right )-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{5 b}\right )-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {3}{2} a \left (\frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {a^2 \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b}+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}\right )-\frac {(a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{5 b}\right )-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}\)

Input:

Int[(a + b*x)^4*Sqrt[a^2 - b^2*x^2],x]
 

Output:

-1/6*((a + b*x)^3*(a^2 - b^2*x^2)^(3/2))/b + (3*a*(-1/5*((a + b*x)^2*(a^2 
- b^2*x^2)^(3/2))/b + (7*a*(-1/4*((a + b*x)*(a^2 - b^2*x^2)^(3/2))/b + (5* 
a*(-1/3*(a^2 - b^2*x^2)^(3/2)/b + a*((x*Sqrt[a^2 - b^2*x^2])/2 + (a^2*ArcT 
an[(b*x)/Sqrt[a^2 - b^2*x^2]])/(2*b))))/4))/5))/2
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 469

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* 
((n + p)/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr 
eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* 
p + 1, 0] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.70

method	result	size
risch	\(-\frac {\left (-40 b^{5} x^{5}-192 a \,b^{4} x^{4}-350 a^{2} b^{3} x^{3}-256 a^{3} b^{2} x^{2}+75 a^{4} b x +448 a^{5}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{240 b}+\frac {21 a^{6} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{16 \sqrt {b^{2}}}\)	\(105\)
default	\(a^{4} \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )+b^{4} \left (-\frac {x^{3} \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{6 b^{2}}+\frac {a^{2} \left (-\frac {x \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{4 b^{2}}+\frac {a^{2} \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{2 b^{2}}\right )+4 a \,b^{3} \left (-\frac {x^{2} \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{5 b^{2}}-\frac {2 a^{2} \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{15 b^{4}}\right )+6 a^{2} b^{2} \left (-\frac {x \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{4 b^{2}}+\frac {a^{2} \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )}{4 b^{2}}\right )-\frac {4 a^{3} \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{3 b}\)	\(330\)

Input:

int((b*x+a)^4*(-b^2*x^2+a^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/240*(-40*b^5*x^5-192*a*b^4*x^4-350*a^2*b^3*x^3-256*a^3*b^2*x^2+75*a^4*b 
*x+448*a^5)/b*(-b^2*x^2+a^2)^(1/2)+21/16*a^6/(b^2)^(1/2)*arctan((b^2)^(1/2 
)*x/(-b^2*x^2+a^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.71 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=-\frac {630 \, a^{6} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) - {\left (40 \, b^{5} x^{5} + 192 \, a b^{4} x^{4} + 350 \, a^{2} b^{3} x^{3} + 256 \, a^{3} b^{2} x^{2} - 75 \, a^{4} b x - 448 \, a^{5}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{240 \, b} \] Input:

integrate((b*x+a)^4*(-b^2*x^2+a^2)^(1/2),x, algorithm="fricas")
 

Output:

-1/240*(630*a^6*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) - (40*b^5*x^5 + 
192*a*b^4*x^4 + 350*a^2*b^3*x^3 + 256*a^3*b^2*x^2 - 75*a^4*b*x - 448*a^5)* 
sqrt(-b^2*x^2 + a^2))/b
 

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.10 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\begin {cases} \frac {21 a^{6} \left (\begin {cases} \frac {\log {\left (- 2 b^{2} x + 2 \sqrt {- b^{2}} \sqrt {a^{2} - b^{2} x^{2}} \right )}}{\sqrt {- b^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- b^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16} + \sqrt {a^{2} - b^{2} x^{2}} \left (- \frac {28 a^{5}}{15 b} - \frac {5 a^{4} x}{16} + \frac {16 a^{3} b x^{2}}{15} + \frac {35 a^{2} b^{2} x^{3}}{24} + \frac {4 a b^{3} x^{4}}{5} + \frac {b^{4} x^{5}}{6}\right ) & \text {for}\: b^{2} \neq 0 \\\sqrt {a^{2}} \left (\begin {cases} a^{4} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{5}}{5 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((b*x+a)**4*(-b**2*x**2+a**2)**(1/2),x)
 

Output:

Piecewise((21*a**6*Piecewise((log(-2*b**2*x + 2*sqrt(-b**2)*sqrt(a**2 - b* 
*2*x**2))/sqrt(-b**2), Ne(a**2, 0)), (x*log(x)/sqrt(-b**2*x**2), True))/16 
 + sqrt(a**2 - b**2*x**2)*(-28*a**5/(15*b) - 5*a**4*x/16 + 16*a**3*b*x**2/ 
15 + 35*a**2*b**2*x**3/24 + 4*a*b**3*x**4/5 + b**4*x**5/6), Ne(b**2, 0)), 
(sqrt(a**2)*Piecewise((a**4*x, Eq(b, 0)), ((a + b*x)**5/(5*b), True)), Tru 
e))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.81 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=-\frac {1}{6} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} b^{2} x^{3} + \frac {21 \, a^{6} \arcsin \left (\frac {b x}{a}\right )}{16 \, b} + \frac {21}{16} \, \sqrt {-b^{2} x^{2} + a^{2}} a^{4} x - \frac {4}{5} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} a b x^{2} - \frac {13}{8} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} a^{2} x - \frac {28 \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} a^{3}}{15 \, b} \] Input:

integrate((b*x+a)^4*(-b^2*x^2+a^2)^(1/2),x, algorithm="maxima")
 

Output:

-1/6*(-b^2*x^2 + a^2)^(3/2)*b^2*x^3 + 21/16*a^6*arcsin(b*x/a)/b + 21/16*sq 
rt(-b^2*x^2 + a^2)*a^4*x - 4/5*(-b^2*x^2 + a^2)^(3/2)*a*b*x^2 - 13/8*(-b^2 
*x^2 + a^2)^(3/2)*a^2*x - 28/15*(-b^2*x^2 + a^2)^(3/2)*a^3/b
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.61 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\frac {21 \, a^{6} \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (b\right )}{16 \, {\left | b \right |}} - \frac {1}{240} \, {\left (\frac {448 \, a^{5}}{b} + {\left (75 \, a^{4} - 2 \, {\left (128 \, a^{3} b + {\left (175 \, a^{2} b^{2} + 4 \, {\left (5 \, b^{4} x + 24 \, a b^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-b^{2} x^{2} + a^{2}} \] Input:

integrate((b*x+a)^4*(-b^2*x^2+a^2)^(1/2),x, algorithm="giac")
 

Output:

21/16*a^6*arcsin(b*x/a)*sgn(a)*sgn(b)/abs(b) - 1/240*(448*a^5/b + (75*a^4 
- 2*(128*a^3*b + (175*a^2*b^2 + 4*(5*b^4*x + 24*a*b^3)*x)*x)*x)*x)*sqrt(-b 
^2*x^2 + a^2)
 

Mupad [F(-1)]

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

int((a^2 - b^2*x^2)^(1/2)*(a + b*x)^4,x)
 

Output:

int((a^2 - b^2*x^2)^(1/2)*(a + b*x)^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\frac {315 \mathit {asin} \left (\frac {b x}{a}\right ) a^{6}-448 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{5}-75 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{4} b x +256 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{3} b^{2} x^{2}+350 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{2} b^{3} x^{3}+192 \sqrt {-b^{2} x^{2}+a^{2}}\, a \,b^{4} x^{4}+40 \sqrt {-b^{2} x^{2}+a^{2}}\, b^{5} x^{5}+448 a^{6}}{240 b} \] Input:

int((b*x+a)^4*(-b^2*x^2+a^2)^(1/2),x)
 

Output:

(315*asin((b*x)/a)*a**6 - 448*sqrt(a**2 - b**2*x**2)*a**5 - 75*sqrt(a**2 - 
 b**2*x**2)*a**4*b*x + 256*sqrt(a**2 - b**2*x**2)*a**3*b**2*x**2 + 350*sqr 
t(a**2 - b**2*x**2)*a**2*b**3*x**3 + 192*sqrt(a**2 - b**2*x**2)*a*b**4*x** 
4 + 40*sqrt(a**2 - b**2*x**2)*b**5*x**5 + 448*a**6)/(240*b)