Integrand size = 28, antiderivative size = 113 \[ \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\frac {(c+d x)^3}{b d^2 \sqrt {b c^2-b d^2 x^2}}+\frac {3 (4 c+d x) \sqrt {b c^2-b d^2 x^2}}{2 b^2 d^2}-\frac {9 c^2 \arctan \left (\frac {\sqrt {b} d x}{\sqrt {b c^2-b d^2 x^2}}\right )}{2 b^{3/2} d^2} \] Output:
(d*x+c)^3/b/d^2/(-b*d^2*x^2+b*c^2)^(1/2)+3/2*(d*x+4*c)*(-b*d^2*x^2+b*c^2)^ (1/2)/b^2/d^2-9/2*c^2*arctan(b^(1/2)*d*x/(-b*d^2*x^2+b*c^2)^(1/2))/b^(3/2) /d^2
Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.98 \[ \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\frac {14 c^3+9 c^2 d x-6 c d^2 x^2-d^3 x^3+18 c^2 \sqrt {c^2-d^2 x^2} \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{2 b d^2 \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:
Integrate[(x*(c + d*x)^3)/(b*c^2 - b*d^2*x^2)^(3/2),x]
Output:
(14*c^3 + 9*c^2*d*x - 6*c*d^2*x^2 - d^3*x^3 + 18*c^2*Sqrt[c^2 - d^2*x^2]*A rcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/(2*b*d^2*Sqrt[b*(c^2 - d^2 *x^2)])
Time = 0.51 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {527, 2346, 27, 455, 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.
\(\displaystyle \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 527 |
\(\displaystyle \frac {4 c^2 (c+d x)}{b d^2 \sqrt {b c^2-b d^2 x^2}}-\frac {\int \frac {4 c^2+3 d x c+d^2 x^2}{\sqrt {b c^2-b d^2 x^2}}dx}{b d}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {4 c^2 (c+d x)}{b d^2 \sqrt {b c^2-b d^2 x^2}}-\frac {-\frac {\int -\frac {3 b c d^2 (3 c+2 d x)}{\sqrt {b c^2-b d^2 x^2}}dx}{2 b d^2}-\frac {x \sqrt {b c^2-b d^2 x^2}}{2 b}}{b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 c^2 (c+d x)}{b d^2 \sqrt {b c^2-b d^2 x^2}}-\frac {\frac {3}{2} c \int \frac {3 c+2 d x}{\sqrt {b c^2-b d^2 x^2}}dx-\frac {x \sqrt {b c^2-b d^2 x^2}}{2 b}}{b d}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {4 c^2 (c+d x)}{b d^2 \sqrt {b c^2-b d^2 x^2}}-\frac {\frac {3}{2} c \left (3 c \int \frac {1}{\sqrt {b c^2-b d^2 x^2}}dx-\frac {2 \sqrt {b c^2-b d^2 x^2}}{b d}\right )-\frac {x \sqrt {b c^2-b d^2 x^2}}{2 b}}{b d}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {4 c^2 (c+d x)}{b d^2 \sqrt {b c^2-b d^2 x^2}}-\frac {\frac {3}{2} c \left (3 c \int \frac {1}{\frac {b d^2 x^2}{b c^2-b d^2 x^2}+1}d\frac {x}{\sqrt {b c^2-b d^2 x^2}}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{b d}\right )-\frac {x \sqrt {b c^2-b d^2 x^2}}{2 b}}{b d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {4 c^2 (c+d x)}{b d^2 \sqrt {b c^2-b d^2 x^2}}-\frac {\frac {3}{2} c \left (\frac {3 c \arctan \left (\frac {\sqrt {b} d x}{\sqrt {b c^2-b d^2 x^2}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{b d}\right )-\frac {x \sqrt {b c^2-b d^2 x^2}}{2 b}}{b d}\) |
Input:
Int[(x*(c + d*x)^3)/(b*c^2 - b*d^2*x^2)^(3/2),x]
Output:
(4*c^2*(c + d*x))/(b*d^2*Sqrt[b*c^2 - b*d^2*x^2]) - (-1/2*(x*Sqrt[b*c^2 - b*d^2*x^2])/b + (3*c*((-2*Sqrt[b*c^2 - b*d^2*x^2])/(b*d) + (3*c*ArcTan[(Sq rt[b]*d*x)/Sqrt[b*c^2 - b*d^2*x^2]])/(Sqrt[b]*d)))/2)/(b*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_S ymbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b*x^2])), x] + Simp[1/(b*d^(m - 2)) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[( 2^(n - 1)*c^(m + n - 1) - d^m*x^m*(c + d*x)^(n - 1))/(c - d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.30
method | result | size |
risch | \(\frac {\left (d x +6 c \right ) \left (-d^{2} x^{2}+c^{2}\right )}{2 d^{2} \sqrt {-b \left (d^{2} x^{2}-c^{2}\right )}\, b}+\frac {-\frac {9 c^{2} \arctan \left (\frac {\sqrt {b \,d^{2}}\, x}{\sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}\right )}{2 d \sqrt {b \,d^{2}}}-\frac {4 c^{2} \sqrt {-b \,d^{2} \left (x -\frac {c}{d}\right )^{2}-2 d b c \left (x -\frac {c}{d}\right )}}{d^{3} b \left (x -\frac {c}{d}\right )}}{b}\) | \(147\) |
default | \(\frac {c^{3}}{b \,d^{2} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}+d^{3} \left (-\frac {x^{3}}{2 b \,d^{2} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}+\frac {3 c^{2} \left (\frac {x}{b \,d^{2} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}-\frac {\arctan \left (\frac {\sqrt {b \,d^{2}}\, x}{\sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}\right )}{b \,d^{2} \sqrt {b \,d^{2}}}\right )}{2 d^{2}}\right )+3 c \,d^{2} \left (-\frac {x^{2}}{b \,d^{2} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}+\frac {2 c^{2}}{b \,d^{4} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}\right )+3 c^{2} d \left (\frac {x}{b \,d^{2} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}-\frac {\arctan \left (\frac {\sqrt {b \,d^{2}}\, x}{\sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}\right )}{b \,d^{2} \sqrt {b \,d^{2}}}\right )\) | \(275\) |
Input:
int(x*(d*x+c)^3/(-b*d^2*x^2+b*c^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*(d*x+6*c)/d^2*(-d^2*x^2+c^2)/(-b*(d^2*x^2-c^2))^(1/2)/b+(-9/2*c^2/d/(b *d^2)^(1/2)*arctan((b*d^2)^(1/2)*x/(-b*d^2*x^2+b*c^2)^(1/2))-4*c^2/d^3/b/( x-c/d)*(-b*d^2*(x-c/d)^2-2*d*b*c*(x-c/d))^(1/2))/b
Time = 0.14 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.12 \[ \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\left [-\frac {9 \, {\left (c^{2} d x - c^{3}\right )} \sqrt {-b} \log \left (2 \, b d^{2} x^{2} - b c^{2} + 2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {-b} d x\right ) - 2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (d^{2} x^{2} + 5 \, c d x - 14 \, c^{2}\right )}}{4 \, {\left (b^{2} d^{3} x - b^{2} c d^{2}\right )}}, \frac {9 \, {\left (c^{2} d x - c^{3}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {b} d x}{b d^{2} x^{2} - b c^{2}}\right ) + \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (d^{2} x^{2} + 5 \, c d x - 14 \, c^{2}\right )}}{2 \, {\left (b^{2} d^{3} x - b^{2} c d^{2}\right )}}\right ] \] Input:
integrate(x*(d*x+c)^3/(-b*d^2*x^2+b*c^2)^(3/2),x, algorithm="fricas")
Output:
[-1/4*(9*(c^2*d*x - c^3)*sqrt(-b)*log(2*b*d^2*x^2 - b*c^2 + 2*sqrt(-b*d^2* x^2 + b*c^2)*sqrt(-b)*d*x) - 2*sqrt(-b*d^2*x^2 + b*c^2)*(d^2*x^2 + 5*c*d*x - 14*c^2))/(b^2*d^3*x - b^2*c*d^2), 1/2*(9*(c^2*d*x - c^3)*sqrt(b)*arctan (sqrt(-b*d^2*x^2 + b*c^2)*sqrt(b)*d*x/(b*d^2*x^2 - b*c^2)) + sqrt(-b*d^2*x ^2 + b*c^2)*(d^2*x^2 + 5*c*d*x - 14*c^2))/(b^2*d^3*x - b^2*c*d^2)]
\[ \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\int \frac {x \left (c + d x\right )^{3}}{\left (- b \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x*(d*x+c)**3/(-b*d**2*x**2+b*c**2)**(3/2),x)
Output:
Integral(x*(c + d*x)**3/(-b*(-c + d*x)*(c + d*x))**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.13 \[ \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=-\frac {d x^{3}}{2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} b} - \frac {3 \, c x^{2}}{\sqrt {-b d^{2} x^{2} + b c^{2}} b} + \frac {9 \, c^{2} x}{2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} b d} + \frac {7 \, c^{3}}{\sqrt {-b d^{2} x^{2} + b c^{2}} b d^{2}} - \frac {9 \, c^{2} \arcsin \left (\frac {d x}{c}\right )}{2 \, b^{\frac {3}{2}} d^{2}} \] Input:
integrate(x*(d*x+c)^3/(-b*d^2*x^2+b*c^2)^(3/2),x, algorithm="maxima")
Output:
-1/2*d*x^3/(sqrt(-b*d^2*x^2 + b*c^2)*b) - 3*c*x^2/(sqrt(-b*d^2*x^2 + b*c^2 )*b) + 9/2*c^2*x/(sqrt(-b*d^2*x^2 + b*c^2)*b*d) + 7*c^3/(sqrt(-b*d^2*x^2 + b*c^2)*b*d^2) - 9/2*c^2*arcsin(d*x/c)/(b^(3/2)*d^2)
Exception generated. \[ \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(d*x+c)^3/(-b*d^2*x^2+b*c^2)^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\int \frac {x\,{\left (c+d\,x\right )}^3}{{\left (b\,c^2-b\,d^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((x*(c + d*x)^3)/(b*c^2 - b*d^2*x^2)^(3/2),x)
Output:
int((x*(c + d*x)^3)/(b*c^2 - b*d^2*x^2)^(3/2), x)
Time = 0.20 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.50 \[ \int \frac {x (c+d x)^3}{\left (b c^2-b d^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {b}\, \left (-9 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) c^{2}+9 \mathit {asin} \left (\frac {d x}{c}\right ) c^{3}-9 \mathit {asin} \left (\frac {d x}{c}\right ) c^{2} d x -18 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c d x +\sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}+18 c^{3}+5 c^{2} d x -6 c \,d^{2} x^{2}-d^{3} x^{3}\right )}{2 b^{2} d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}-c +d x \right )} \] Input:
int(x*(d*x+c)^3/(-b*d^2*x^2+b*c^2)^(3/2),x)
Output:
(sqrt(b)*( - 9*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*c**2 + 9*asin((d*x)/c) *c**3 - 9*asin((d*x)/c)*c**2*d*x - 18*sqrt(c**2 - d**2*x**2)*c**2 + 5*sqrt (c**2 - d**2*x**2)*c*d*x + sqrt(c**2 - d**2*x**2)*d**2*x**2 + 18*c**3 + 5* c**2*d*x - 6*c*d**2*x**2 - d**3*x**3))/(2*b**2*d**2*(sqrt(c**2 - d**2*x**2 ) - c + d*x))