Integrand size = 36, antiderivative size = 146 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 \sqrt {b c^2-b d^2 x^2}}{5 b e (e x)^{5/2} \sqrt {c+d x}}-\frac {6 d \sqrt {b c^2-b d^2 x^2}}{5 b c e^2 (e x)^{3/2} \sqrt {c+d x}}-\frac {12 d^2 \sqrt {b c^2-b d^2 x^2}}{5 b c^2 e^3 \sqrt {e x} \sqrt {c+d x}} \] Output:
-2/5*(-b*d^2*x^2+b*c^2)^(1/2)/b/e/(e*x)^(5/2)/(d*x+c)^(1/2)-6/5*d*(-b*d^2* x^2+b*c^2)^(1/2)/b/c/e^2/(e*x)^(3/2)/(d*x+c)^(1/2)-12/5*d^2*(-b*d^2*x^2+b* c^2)^(1/2)/b/c^2/e^3/(e*x)^(1/2)/(d*x+c)^(1/2)
Time = 5.84 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.45 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 x (c-d x) \sqrt {c+d x} \left (c^2+3 c d x+6 d^2 x^2\right )}{5 c^2 (e x)^{7/2} \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:
Integrate[(c + d*x)^(3/2)/((e*x)^(7/2)*Sqrt[b*c^2 - b*d^2*x^2]),x]
Output:
(-2*x*(c - d*x)*Sqrt[c + d*x]*(c^2 + 3*c*d*x + 6*d^2*x^2))/(5*c^2*(e*x)^(7 /2)*Sqrt[b*(c^2 - d^2*x^2)])
Time = 0.47 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {580, 579, 579}
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.
\(\displaystyle \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx\) |
\(\Big \downarrow \) 580 |
\(\displaystyle \frac {9 d \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \sqrt {b c^2-b d^2 x^2}}dx}{5 e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{5 b e (e x)^{5/2} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 579 |
\(\displaystyle \frac {9 d \left (\frac {2 d \int \frac {\sqrt {c+d x}}{(e x)^{3/2} \sqrt {b c^2-b d^2 x^2}}dx}{3 c e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{3 b c e (e x)^{3/2} \sqrt {c+d x}}\right )}{5 e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{5 b e (e x)^{5/2} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 579 |
\(\displaystyle \frac {9 d \left (-\frac {4 d \sqrt {b c^2-b d^2 x^2}}{3 b c^2 e^2 \sqrt {e x} \sqrt {c+d x}}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{3 b c e (e x)^{3/2} \sqrt {c+d x}}\right )}{5 e}-\frac {2 \sqrt {b c^2-b d^2 x^2}}{5 b e (e x)^{5/2} \sqrt {c+d x}}\) |
Input:
Int[(c + d*x)^(3/2)/((e*x)^(7/2)*Sqrt[b*c^2 - b*d^2*x^2]),x]
Output:
(-2*Sqrt[b*c^2 - b*d^2*x^2])/(5*b*e*(e*x)^(5/2)*Sqrt[c + d*x]) + (9*d*((-2 *Sqrt[b*c^2 - b*d^2*x^2])/(3*b*c*e*(e*x)^(3/2)*Sqrt[c + d*x]) - (4*d*Sqrt[ b*c^2 - b*d^2*x^2])/(3*b*c^2*e^2*Sqrt[e*x]*Sqrt[c + d*x])))/(5*e)
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*c*e*(m + 1))), x] - Simp[d*((n - m - 2)/(c*e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0] && LtQ[m, -1] && (IntegerQ[2*p] | | IntegerQ[m])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 2)*((a + b*x^2)^(p + 1)/(b*e*(m + 1))), x] + Simp[d*((2*m + p + 3)/(e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p - 1, 0] && LtQ[m, -1] && IntegerQ [p + 1/2]
Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.40
method | result | size |
gosper | \(-\frac {2 \left (-d x +c \right ) x \left (6 d^{2} x^{2}+3 c d x +c^{2}\right ) \sqrt {d x +c}}{5 c^{2} \left (e x \right )^{\frac {7}{2}} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}\) | \(59\) |
orering | \(-\frac {2 \left (-d x +c \right ) x \left (6 d^{2} x^{2}+3 c d x +c^{2}\right ) \sqrt {d x +c}}{5 c^{2} \left (e x \right )^{\frac {7}{2}} \sqrt {-b \,x^{2} d^{2}+b \,c^{2}}}\) | \(59\) |
default | \(-\frac {2 \sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (6 d^{2} x^{2}+3 c d x +c^{2}\right )}{5 b \,x^{2} c^{2} \sqrt {d x +c}\, e^{3} \sqrt {e x}}\) | \(60\) |
Input:
int((d*x+c)^(3/2)/(e*x)^(7/2)/(-b*d^2*x^2+b*c^2)^(1/2),x,method=_RETURNVER BOSE)
Output:
-2/5*(-d*x+c)*x*(6*d^2*x^2+3*c*d*x+c^2)*(d*x+c)^(1/2)/c^2/(e*x)^(7/2)/(-b* d^2*x^2+b*c^2)^(1/2)
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.51 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (6 \, d^{2} x^{2} + 3 \, c d x + c^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{5 \, {\left (b c^{2} d e^{4} x^{4} + b c^{3} e^{4} x^{3}\right )}} \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(7/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm= "fricas")
Output:
-2/5*sqrt(-b*d^2*x^2 + b*c^2)*(6*d^2*x^2 + 3*c*d*x + c^2)*sqrt(d*x + c)*sq rt(e*x)/(b*c^2*d*e^4*x^4 + b*c^3*e^4*x^3)
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(3/2)/(e*x)**(7/2)/(-b*d**2*x**2+b*c**2)**(1/2),x)
Output:
Timed out
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{\sqrt {-b d^{2} x^{2} + b c^{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(7/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm= "maxima")
Output:
integrate((d*x + c)^(3/2)/(sqrt(-b*d^2*x^2 + b*c^2)*(e*x)^(7/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (122) = 244\).
Time = 0.22 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.47 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {8 \, {\left (3 \, \sqrt {-b d e} b^{4} c^{3} d^{10} e^{9} {\left | d \right |} {\left | e \right |} - 15 \, \sqrt {-b d e} {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{2} b^{3} c^{2} d^{8} e^{7} {\left | d \right |} {\left | e \right |} + 25 \, \sqrt {-b d e} {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{4} b^{2} c d^{6} e^{5} {\left | d \right |} {\left | e \right |} - 5 \, \sqrt {-b d e} {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{6} b d^{4} e^{3} {\left | d \right |} {\left | e \right |}\right )} d^{2}}{5 \, {\left (b c d^{2} e^{2} - {\left (\sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {-b d e} - \sqrt {b c d^{2} e^{2} - {\left ({\left (d x + c\right )} d e - c d e\right )} b d e}\right )}^{2}\right )}^{5} e^{4} {\left | d \right |}} \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(7/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm= "giac")
Output:
-8/5*(3*sqrt(-b*d*e)*b^4*c^3*d^10*e^9*abs(d)*abs(e) - 15*sqrt(-b*d*e)*(sqr t((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + c)*d*e - c*d*e)*b*d*e))^2*b^3*c^2*d^8*e^7*abs(d)*abs(e) + 25*sqrt(-b*d*e)*(sqrt(( d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + c)*d*e - c *d*e)*b*d*e))^4*b^2*c*d^6*e^5*abs(d)*abs(e) - 5*sqrt(-b*d*e)*(sqrt((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + c)*d*e - c*d*e)* b*d*e))^6*b*d^4*e^3*abs(d)*abs(e))*d^2/((b*c*d^2*e^2 - (sqrt((d*x + c)*d*e - c*d*e)*sqrt(-b*d*e) - sqrt(b*c*d^2*e^2 - ((d*x + c)*d*e - c*d*e)*b*d*e) )^2)^5*e^4*abs(d))
Time = 7.51 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.71 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\sqrt {b\,c^2-b\,d^2\,x^2}\,\left (\frac {2\,\sqrt {c+d\,x}}{5\,b\,d\,e^3}+\frac {6\,x\,\sqrt {c+d\,x}}{5\,b\,c\,e^3}+\frac {12\,d\,x^2\,\sqrt {c+d\,x}}{5\,b\,c^2\,e^3}\right )}{x^3\,\sqrt {e\,x}+\frac {c\,x^2\,\sqrt {e\,x}}{d}} \] Input:
int((c + d*x)^(3/2)/((e*x)^(7/2)*(b*c^2 - b*d^2*x^2)^(1/2)),x)
Output:
-((b*c^2 - b*d^2*x^2)^(1/2)*((2*(c + d*x)^(1/2))/(5*b*d*e^3) + (6*x*(c + d *x)^(1/2))/(5*b*c*e^3) + (12*d*x^2*(c + d*x)^(1/2))/(5*b*c^2*e^3)))/(x^3*( e*x)^(1/2) + (c*x^2*(e*x)^(1/2))/d)
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {2 \sqrt {e}\, \sqrt {b}\, \left (-\sqrt {x}\, \sqrt {-d x +c}\, c^{2}-3 \sqrt {x}\, \sqrt {-d x +c}\, c d x -6 \sqrt {x}\, \sqrt {-d x +c}\, d^{2} x^{2}+6 \sqrt {d}\, d^{2} i \,x^{3}\right )}{5 b \,c^{2} e^{4} x^{3}} \] Input:
int((d*x+c)^(3/2)/(e*x)^(7/2)/(-b*d^2*x^2+b*c^2)^(1/2),x)
Output:
(2*sqrt(e)*sqrt(b)*( - sqrt(x)*sqrt(c - d*x)*c**2 - 3*sqrt(x)*sqrt(c - d*x )*c*d*x - 6*sqrt(x)*sqrt(c - d*x)*d**2*x**2 + 6*sqrt(d)*d**2*i*x**3))/(5*b *c**2*e**4*x**3)