Integrand size = 24, antiderivative size = 131 \[ \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx=\frac {7}{16} a^4 x \sqrt {a^2-b^2 x^2}+\frac {7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac {7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac {7 a^6 \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{16 b} \]
7/24*a^2*x*(-b^2*x^2+a^2)^(3/2)-7/30*a*(-b^2*x^2+a^2)^(5/2)/b-1/6*(b*x+a)* (-b^2*x^2+a^2)^(5/2)/b+7/16*a^6*arctan(b*x/(-b^2*x^2+a^2)^(1/2))/b+7/16*a^ 4*x*(-b^2*x^2+a^2)^(1/2)
Time = 0.62 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.87 \[ \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx=\frac {\sqrt {a^2-b^2 x^2} \left (-96 a^5+135 a^4 b x+192 a^3 b^2 x^2+10 a^2 b^3 x^3-96 a b^4 x^4-40 b^5 x^5\right )-210 a^6 \arctan \left (\frac {b x}{\sqrt {a^2}-\sqrt {a^2-b^2 x^2}}\right )}{240 b} \]
(Sqrt[a^2 - b^2*x^2]*(-96*a^5 + 135*a^4*b*x + 192*a^3*b^2*x^2 + 10*a^2*b^3 *x^3 - 96*a*b^4*x^4 - 40*b^5*x^5) - 210*a^6*ArcTan[(b*x)/(Sqrt[a^2] - Sqrt [a^2 - b^2*x^2])])/(240*b)
Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {469, 455, 211, 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.
\(\displaystyle \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle \frac {7}{6} a \int (a+b x) \left (a^2-b^2 x^2\right )^{3/2}dx-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {7}{6} a \left (a \int \left (a^2-b^2 x^2\right )^{3/2}dx-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{5 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {7}{6} a \left (a \left (\frac {3}{4} a^2 \int \sqrt {a^2-b^2 x^2}dx+\frac {1}{4} x \left (a^2-b^2 x^2\right )^{3/2}\right )-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{5 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {7}{6} a \left (a \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {1}{\sqrt {a^2-b^2 x^2}}dx+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )+\frac {1}{4} x \left (a^2-b^2 x^2\right )^{3/2}\right )-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{5 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {7}{6} a \left (a \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {1}{\frac {b^2 x^2}{a^2-b^2 x^2}+1}d\frac {x}{\sqrt {a^2-b^2 x^2}}+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )+\frac {1}{4} x \left (a^2-b^2 x^2\right )^{3/2}\right )-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{5 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {7}{6} a \left (a \left (\frac {3}{4} a^2 \left (\frac {a^2 \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b}+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )+\frac {1}{4} x \left (a^2-b^2 x^2\right )^{3/2}\right )-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{5 b}\right )-\frac {(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}\) |
-1/6*((a + b*x)*(a^2 - b^2*x^2)^(5/2))/b + (7*a*(-1/5*(a^2 - b^2*x^2)^(5/2 )/b + a*((x*(a^2 - b^2*x^2)^(3/2))/4 + (3*a^2*((x*Sqrt[a^2 - b^2*x^2])/2 + (a^2*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(2*b)))/4)))/6
3.8.89.3.1 Defintions of rubi rules used
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])
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])
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]
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]
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]
Time = 2.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\left (40 b^{5} x^{5}+96 a \,b^{4} x^{4}-10 a^{2} b^{3} x^{3}-192 a^{3} b^{2} x^{2}-135 a^{4} b x +96 a^{5}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{240 b}+\frac {7 a^{6} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{16 \sqrt {b^{2}}}\) | \(105\) |
default | \(a^{2} \left (\frac {x \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )}{4}\right )+b^{2} \left (-\frac {x \left (-b^{2} x^{2}+a^{2}\right )^{\frac {5}{2}}}{6 b^{2}}+\frac {a^{2} \left (\frac {x \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )}{4}\right )}{6 b^{2}}\right )-\frac {2 a \left (-b^{2} x^{2}+a^{2}\right )^{\frac {5}{2}}}{5 b}\) | \(205\) |
-1/240*(40*b^5*x^5+96*a*b^4*x^4-10*a^2*b^3*x^3-192*a^3*b^2*x^2-135*a^4*b*x +96*a^5)/b*(-b^2*x^2+a^2)^(1/2)+7/16*a^6/(b^2)^(1/2)*arctan((b^2)^(1/2)*x/ (-b^2*x^2+a^2)^(1/2))
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80 \[ \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx=-\frac {210 \, a^{6} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) + {\left (40 \, b^{5} x^{5} + 96 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} - 192 \, a^{3} b^{2} x^{2} - 135 \, a^{4} b x + 96 \, a^{5}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{240 \, b} \]
-1/240*(210*a^6*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) + (40*b^5*x^5 + 96*a*b^4*x^4 - 10*a^2*b^3*x^3 - 192*a^3*b^2*x^2 - 135*a^4*b*x + 96*a^5)*sq rt(-b^2*x^2 + a^2))/b
Time = 0.58 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.24 \[ \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {7 a^{6} \left (\begin {cases} \frac {\log {\left (- 2 b^{2} x + 2 \sqrt {- b^{2}} \sqrt {a^{2} - b^{2} x^{2}} \right )}}{\sqrt {- b^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- b^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16} + \sqrt {a^{2} - b^{2} x^{2}} \left (- \frac {2 a^{5}}{5 b} + \frac {9 a^{4} x}{16} + \frac {4 a^{3} b x^{2}}{5} + \frac {a^{2} b^{2} x^{3}}{24} - \frac {2 a b^{3} x^{4}}{5} - \frac {b^{4} x^{5}}{6}\right ) & \text {for}\: b^{2} \neq 0 \\\left (a^{2}\right )^{\frac {3}{2}} \left (\begin {cases} a^{2} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{3}}{3 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((7*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)*(-2*a**5/(5*b) + 9*a**4*x/16 + 4*a**3*b*x**2/5 + a**2*b**2*x**3/24 - 2*a*b**3*x**4/5 - b**4*x**5/6), Ne(b**2, 0)), ((a**2)* *(3/2)*Piecewise((a**2*x, Eq(b, 0)), ((a + b*x)**3/(3*b), True)), True))
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.71 \[ \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx=\frac {7 \, a^{6} \arcsin \left (\frac {b x}{a}\right )}{16 \, b} + \frac {7}{16} \, \sqrt {-b^{2} x^{2} + a^{2}} a^{4} x + \frac {7}{24} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} a^{2} x - \frac {1}{6} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {5}{2}} x - \frac {2 \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {5}{2}} a}{5 \, b} \]
7/16*a^6*arcsin(b*x/a)/b + 7/16*sqrt(-b^2*x^2 + a^2)*a^4*x + 7/24*(-b^2*x^ 2 + a^2)^(3/2)*a^2*x - 1/6*(-b^2*x^2 + a^2)^(5/2)*x - 2/5*(-b^2*x^2 + a^2) ^(5/2)*a/b
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.70 \[ \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx=\frac {7 \, a^{6} \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (b\right )}{16 \, {\left | b \right |}} - \frac {1}{240} \, {\left (\frac {96 \, a^{5}}{b} - {\left (135 \, a^{4} + 2 \, {\left (96 \, a^{3} b + {\left (5 \, a^{2} b^{2} - 4 \, {\left (5 \, b^{4} x + 12 \, a b^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-b^{2} x^{2} + a^{2}} \]
7/16*a^6*arcsin(b*x/a)*sgn(a)*sgn(b)/abs(b) - 1/240*(96*a^5/b - (135*a^4 + 2*(96*a^3*b + (5*a^2*b^2 - 4*(5*b^4*x + 12*a*b^3)*x)*x)*x)*x)*sqrt(-b^2*x ^2 + a^2)
Timed out. \[ \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx=\int {\left (a^2-b^2\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2 \,d x \]