Integrand size = 31, antiderivative size = 115 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\frac {8 c (B c-9 A d) \left (c^2-d^2 x^2\right )^{5/2}}{315 d^2 (c+d x)^{5/2}}+\frac {2 (B c-9 A d) \left (c^2-d^2 x^2\right )^{5/2}}{63 d^2 (c+d x)^{3/2}}-\frac {2 B \left (c^2-d^2 x^2\right )^{5/2}}{9 d^2 \sqrt {c+d x}} \] Output:
8/315*c*(-9*A*d+B*c)*(-d^2*x^2+c^2)^(5/2)/d^2/(d*x+c)^(5/2)+2/63*(-9*A*d+B *c)*(-d^2*x^2+c^2)^(5/2)/d^2/(d*x+c)^(3/2)-2/9*B*(-d^2*x^2+c^2)^(5/2)/d^2/ (d*x+c)^(1/2)
Time = 0.65 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=-\frac {2 (c-d x)^2 \sqrt {c^2-d^2 x^2} \left (9 A d (9 c+5 d x)+B \left (26 c^2+65 c d x+35 d^2 x^2\right )\right )}{315 d^2 \sqrt {c+d x}} \] Input:
Integrate[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/Sqrt[c + d*x],x]
Output:
(-2*(c - d*x)^2*Sqrt[c^2 - d^2*x^2]*(9*A*d*(9*c + 5*d*x) + B*(26*c^2 + 65* c*d*x + 35*d^2*x^2)))/(315*d^2*Sqrt[c + d*x])
Time = 0.38 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {672, 459, 458}
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.
\(\displaystyle \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 672 |
\(\displaystyle -\frac {(B c-9 A d) \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}}dx}{9 d}-\frac {2 B \left (c^2-d^2 x^2\right )^{5/2}}{9 d^2 \sqrt {c+d x}}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle -\frac {(B c-9 A d) \left (\frac {4}{7} c \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{3/2}}dx-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d (c+d x)^{3/2}}\right )}{9 d}-\frac {2 B \left (c^2-d^2 x^2\right )^{5/2}}{9 d^2 \sqrt {c+d x}}\) |
\(\Big \downarrow \) 458 |
\(\displaystyle -\frac {\left (-\frac {2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d (c+d x)^{3/2}}-\frac {8 c \left (c^2-d^2 x^2\right )^{5/2}}{35 d (c+d x)^{5/2}}\right ) (B c-9 A d)}{9 d}-\frac {2 B \left (c^2-d^2 x^2\right )^{5/2}}{9 d^2 \sqrt {c+d x}}\) |
Input:
Int[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/Sqrt[c + d*x],x]
Output:
(-2*B*(c^2 - d^2*x^2)^(5/2))/(9*d^2*Sqrt[c + d*x]) - ((B*c - 9*A*d)*((-8*c *(c^2 - d^2*x^2)^(5/2))/(35*d*(c + d*x)^(5/2)) - (2*(c^2 - d^2*x^2)^(5/2)) /(7*d*(c + d*x)^(3/2))))/(9*d)
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*(p + 1))), x] /; FreeQ[{a, b, c , d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
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* (Simplify[n + p]/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[Simplif y[n + p], 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)) Int[(d + e*x )^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^ 2 + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Time = 0.41 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(-\frac {2 \left (-d x +c \right ) \left (35 B \,d^{2} x^{2}+45 A \,d^{2} x +65 B c d x +81 A c d +26 B \,c^{2}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{315 d^{2} \left (d x +c \right )^{\frac {3}{2}}}\) | \(67\) |
orering | \(-\frac {2 \left (-d x +c \right ) \left (35 B \,d^{2} x^{2}+45 A \,d^{2} x +65 B c d x +81 A c d +26 B \,c^{2}\right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{315 d^{2} \left (d x +c \right )^{\frac {3}{2}}}\) | \(67\) |
default | \(-\frac {2 \sqrt {-d^{2} x^{2}+c^{2}}\, \left (-d x +c \right )^{2} \left (35 B \,d^{2} x^{2}+45 A \,d^{2} x +65 B c d x +81 A c d +26 B \,c^{2}\right )}{315 \sqrt {d x +c}\, d^{2}}\) | \(69\) |
risch | \(-\frac {2 \sqrt {\frac {-d^{2} x^{2}+c^{2}}{d x +c}}\, \sqrt {d x +c}\, \left (35 B \,d^{4} x^{4}+45 A \,d^{4} x^{3}-5 B c \,d^{3} x^{3}-9 A c \,d^{3} x^{2}-69 x^{2} c^{2} B \,d^{2}-117 A \,c^{2} d^{2} x +13 B \,c^{3} d x +81 A \,c^{3} d +26 B \,c^{4}\right ) \sqrt {-d x +c}}{315 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2}}\) | \(139\) |
Input:
int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/315*(-d*x+c)*(35*B*d^2*x^2+45*A*d^2*x+65*B*c*d*x+81*A*c*d+26*B*c^2)*(-d ^2*x^2+c^2)^(3/2)/d^2/(d*x+c)^(3/2)
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=-\frac {2 \, {\left (35 \, B d^{4} x^{4} + 26 \, B c^{4} + 81 \, A c^{3} d - 5 \, {\left (B c d^{3} - 9 \, A d^{4}\right )} x^{3} - 3 \, {\left (23 \, B c^{2} d^{2} + 3 \, A c d^{3}\right )} x^{2} + 13 \, {\left (B c^{3} d - 9 \, A c^{2} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{315 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(1/2),x, algorithm="fricas" )
Output:
-2/315*(35*B*d^4*x^4 + 26*B*c^4 + 81*A*c^3*d - 5*(B*c*d^3 - 9*A*d^4)*x^3 - 3*(23*B*c^2*d^2 + 3*A*c*d^3)*x^2 + 13*(B*c^3*d - 9*A*c^2*d^2)*x)*sqrt(-d^ 2*x^2 + c^2)*sqrt(d*x + c)/(d^3*x + c*d^2)
\[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\sqrt {c + d x}}\, dx \] Input:
integrate((B*x+A)*(-d**2*x**2+c**2)**(3/2)/(d*x+c)**(1/2),x)
Output:
Integral((-(-c + d*x)*(c + d*x))**(3/2)*(A + B*x)/sqrt(c + d*x), x)
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=-\frac {2 \, {\left (5 \, d^{3} x^{3} - c d^{2} x^{2} - 13 \, c^{2} d x + 9 \, c^{3}\right )} \sqrt {-d x + c} A}{35 \, d} - \frac {2 \, {\left (35 \, d^{4} x^{4} - 5 \, c d^{3} x^{3} - 69 \, c^{2} d^{2} x^{2} + 13 \, c^{3} d x + 26 \, c^{4}\right )} \sqrt {-d x + c} B}{315 \, d^{2}} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(1/2),x, algorithm="maxima" )
Output:
-2/35*(5*d^3*x^3 - c*d^2*x^2 - 13*c^2*d*x + 9*c^3)*sqrt(-d*x + c)*A/d - 2/ 315*(35*d^4*x^4 - 5*c*d^3*x^3 - 69*c^2*d^2*x^2 + 13*c^3*d*x + 26*c^4)*sqrt (-d*x + c)*B/d^2
Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.67 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=-\frac {2 \, {\left (105 \, {\left (-d x + c\right )}^{\frac {3}{2}} A c^{2} d - 21 \, {\left (3 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} - 5 \, {\left (-d x + c\right )}^{\frac {3}{2}} c\right )} B c^{2} + 3 \, {\left (15 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} + 42 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c - 35 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{2}\right )} A d + {\left (35 \, {\left (d x - c\right )}^{4} \sqrt {-d x + c} + 135 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} c + 189 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} c^{2} - 105 \, {\left (-d x + c\right )}^{\frac {3}{2}} c^{3}\right )} B\right )}}{315 \, d^{2}} \] Input:
integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
-2/315*(105*(-d*x + c)^(3/2)*A*c^2*d - 21*(3*(d*x - c)^2*sqrt(-d*x + c) - 5*(-d*x + c)^(3/2)*c)*B*c^2 + 3*(15*(d*x - c)^3*sqrt(-d*x + c) + 42*(d*x - c)^2*sqrt(-d*x + c)*c - 35*(-d*x + c)^(3/2)*c^2)*A*d + (35*(d*x - c)^4*sq rt(-d*x + c) + 135*(d*x - c)^3*sqrt(-d*x + c)*c + 189*(d*x - c)^2*sqrt(-d* x + c)*c^2 - 105*(-d*x + c)^(3/2)*c^3)*B)/d^2
Time = 9.67 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=-\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {52\,B\,c^4+162\,A\,d\,c^3}{315\,d^2}-\frac {2\,c\,x^2\,\left (3\,A\,d+23\,B\,c\right )}{105}+\frac {2\,B\,d^2\,x^4}{9}+\frac {x^3\,\left (90\,A\,d^4-10\,B\,c\,d^3\right )}{315\,d^2}-\frac {26\,c^2\,x\,\left (9\,A\,d-B\,c\right )}{315\,d}\right )}{\sqrt {c+d\,x}} \] Input:
int(((c^2 - d^2*x^2)^(3/2)*(A + B*x))/(c + d*x)^(1/2),x)
Output:
-((c^2 - d^2*x^2)^(1/2)*((52*B*c^4 + 162*A*c^3*d)/(315*d^2) - (2*c*x^2*(3* A*d + 23*B*c))/105 + (2*B*d^2*x^4)/9 + (x^3*(90*A*d^4 - 10*B*c*d^3))/(315* d^2) - (26*c^2*x*(9*A*d - B*c))/(315*d)))/(c + d*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {-d x +c}\, \left (-35 b \,d^{4} x^{4}-45 a \,d^{4} x^{3}+5 b c \,d^{3} x^{3}+9 a c \,d^{3} x^{2}+69 b \,c^{2} d^{2} x^{2}+117 a \,c^{2} d^{2} x -13 b \,c^{3} d x -81 a \,c^{3} d -26 b \,c^{4}\right )}{315 d^{2}} \] Input:
int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(1/2),x)
Output:
(2*sqrt(c - d*x)*( - 81*a*c**3*d + 117*a*c**2*d**2*x + 9*a*c*d**3*x**2 - 4 5*a*d**4*x**3 - 26*b*c**4 - 13*b*c**3*d*x + 69*b*c**2*d**2*x**2 + 5*b*c*d* *3*x**3 - 35*b*d**4*x**4))/(315*d**2)