Integrand size = 18, antiderivative size = 92 \[ \int x (c+d x) \sqrt {a+b x^2} \, dx=\frac {a d x \sqrt {a+b x^2}}{8 b}+\frac {1}{4} d x^3 \sqrt {a+b x^2}+\frac {c \left (a+b x^2\right )^{3/2}}{3 b}-\frac {a^2 d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2}} \] Output:
1/8*a*d*x*(b*x^2+a)^(1/2)/b+1/4*d*x^3*(b*x^2+a)^(1/2)+1/3*c*(b*x^2+a)^(3/2 )/b-1/8*a^2*d*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(3/2)
Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.84 \[ \int x (c+d x) \sqrt {a+b x^2} \, dx=\frac {\sqrt {a+b x^2} \left (8 a c+3 a d x+8 b c x^2+6 b d x^3\right )}{24 b}+\frac {a^2 d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{3/2}} \] Input:
Integrate[x*(c + d*x)*Sqrt[a + b*x^2],x]
Output:
(Sqrt[a + b*x^2]*(8*a*c + 3*a*d*x + 8*b*c*x^2 + 6*b*d*x^3))/(24*b) + (a^2* d*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8*b^(3/2))
Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {533, 455, 211, 224, 219}
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.
\(\displaystyle \int x \sqrt {a+b x^2} (c+d x) \, dx\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 b}-\frac {\int (a d-4 b c x) \sqrt {b x^2+a}dx}{4 b}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 b}-\frac {a d \int \sqrt {b x^2+a}dx-\frac {4}{3} c \left (a+b x^2\right )^{3/2}}{4 b}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 b}-\frac {a d \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )-\frac {4}{3} c \left (a+b x^2\right )^{3/2}}{4 b}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 b}-\frac {a d \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )-\frac {4}{3} c \left (a+b x^2\right )^{3/2}}{4 b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {d x \left (a+b x^2\right )^{3/2}}{4 b}-\frac {a d \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )-\frac {4}{3} c \left (a+b x^2\right )^{3/2}}{4 b}\) |
Input:
Int[x*(c + d*x)*Sqrt[a + b*x^2],x]
Output:
(d*x*(a + b*x^2)^(3/2))/(4*b) - ((-4*c*(a + b*x^2)^(3/2))/3 + a*d*((x*Sqrt [a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/(4 *b)
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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {\left (6 b d \,x^{3}+8 b c \,x^{2}+3 a d x +8 a c \right ) \sqrt {b \,x^{2}+a}}{24 b}-\frac {a^{2} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {3}{2}}}\) | \(65\) |
default | \(\frac {c \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}+d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )\) | \(76\) |
Input:
int(x*(d*x+c)*(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/24*(6*b*d*x^3+8*b*c*x^2+3*a*d*x+8*a*c)*(b*x^2+a)^(1/2)/b-1/8*a^2/b^(3/2) *d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))
Time = 0.11 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.71 \[ \int x (c+d x) \sqrt {a+b x^2} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} d \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, b^{2} d x^{3} + 8 \, b^{2} c x^{2} + 3 \, a b d x + 8 \, a b c\right )} \sqrt {b x^{2} + a}}{48 \, b^{2}}, \frac {3 \, a^{2} \sqrt {-b} d \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (6 \, b^{2} d x^{3} + 8 \, b^{2} c x^{2} + 3 \, a b d x + 8 \, a b c\right )} \sqrt {b x^{2} + a}}{24 \, b^{2}}\right ] \] Input:
integrate(x*(d*x+c)*(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
[1/48*(3*a^2*sqrt(b)*d*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2 *(6*b^2*d*x^3 + 8*b^2*c*x^2 + 3*a*b*d*x + 8*a*b*c)*sqrt(b*x^2 + a))/b^2, 1 /24*(3*a^2*sqrt(-b)*d*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (6*b^2*d*x^3 + 8*b^2*c*x^2 + 3*a*b*d*x + 8*a*b*c)*sqrt(b*x^2 + a))/b^2]
Time = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int x (c+d x) \sqrt {a+b x^2} \, dx=\begin {cases} - \frac {a^{2} d \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b} + \sqrt {a + b x^{2}} \left (\frac {a c}{3 b} + \frac {a d x}{8 b} + \frac {c x^{2}}{3} + \frac {d x^{3}}{4}\right ) & \text {for}\: b \neq 0 \\\sqrt {a} \left (\frac {c x^{2}}{2} + \frac {d x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x*(d*x+c)*(b*x**2+a)**(1/2),x)
Output:
Piecewise((-a**2*d*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt (b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True))/(8*b) + sqrt(a + b*x**2)*(a *c/(3*b) + a*d*x/(8*b) + c*x**2/3 + d*x**3/4), Ne(b, 0)), (sqrt(a)*(c*x**2 /2 + d*x**3/3), True))
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.73 \[ \int x (c+d x) \sqrt {a+b x^2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} d x}{4 \, b} - \frac {\sqrt {b x^{2} + a} a d x}{8 \, b} - \frac {a^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} c}{3 \, b} \] Input:
integrate(x*(d*x+c)*(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
1/4*(b*x^2 + a)^(3/2)*d*x/b - 1/8*sqrt(b*x^2 + a)*a*d*x/b - 1/8*a^2*d*arcs inh(b*x/sqrt(a*b))/b^(3/2) + 1/3*(b*x^2 + a)^(3/2)*c/b
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int x (c+d x) \sqrt {a+b x^2} \, dx=\frac {a^{2} d \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {3}{2}}} + \frac {1}{24} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left (3 \, d x + 4 \, c\right )} x + \frac {3 \, a d}{b}\right )} x + \frac {8 \, a c}{b}\right )} \] Input:
integrate(x*(d*x+c)*(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
1/8*a^2*d*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) + 1/24*sqrt(b*x^2 + a)*((2*(3*d*x + 4*c)*x + 3*a*d/b)*x + 8*a*c/b)
Timed out. \[ \int x (c+d x) \sqrt {a+b x^2} \, dx=\int x\,\sqrt {b\,x^2+a}\,\left (c+d\,x\right ) \,d x \] Input:
int(x*(a + b*x^2)^(1/2)*(c + d*x),x)
Output:
int(x*(a + b*x^2)^(1/2)*(c + d*x), x)
Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.02 \[ \int x (c+d x) \sqrt {a+b x^2} \, dx=\frac {8 \sqrt {b \,x^{2}+a}\, a b c +3 \sqrt {b \,x^{2}+a}\, a b d x +8 \sqrt {b \,x^{2}+a}\, b^{2} c \,x^{2}+6 \sqrt {b \,x^{2}+a}\, b^{2} d \,x^{3}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d}{24 b^{2}} \] Input:
int(x*(d*x+c)*(b*x^2+a)^(1/2),x)
Output:
(8*sqrt(a + b*x**2)*a*b*c + 3*sqrt(a + b*x**2)*a*b*d*x + 8*sqrt(a + b*x**2 )*b**2*c*x**2 + 6*sqrt(a + b*x**2)*b**2*d*x**3 - 3*sqrt(b)*log((sqrt(a + b *x**2) + sqrt(b)*x)/sqrt(a))*a**2*d)/(24*b**2)