Optimal. Leaf size=67 \[ \frac {a}{c^2 x \sqrt {d x-c} \sqrt {c+d x}}-\frac {x \left (2 a d^2+b c^2\right )}{c^4 \sqrt {d x-c} \sqrt {c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {454, 39} \[ \frac {a}{c^2 x \sqrt {d x-c} \sqrt {c+d x}}-\frac {x \left (2 a d^2+b c^2\right )}{c^4 \sqrt {d x-c} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 39
Rule 454
Rubi steps
\begin {align*} \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac {a}{c^2 x \sqrt {-c+d x} \sqrt {c+d x}}+\left (b+\frac {2 a d^2}{c^2}\right ) \int \frac {1}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\\ &=\frac {a}{c^2 x \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (b c^2+2 a d^2\right ) x}{c^4 \sqrt {-c+d x} \sqrt {c+d x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 51, normalized size = 0.76 \[ \frac {a \left (c^2-2 d^2 x^2\right )-b c^2 x^2}{c^4 x \sqrt {d x-c} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.00, size = 103, normalized size = 1.54 \[ -\frac {{\left (b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{3} - {\left (a c^{2} d - {\left (b c^{2} d + 2 \, a d^{3}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c} - {\left (b c^{4} + 2 \, a c^{2} d^{2}\right )} x}{c^{4} d^{3} x^{3} - c^{6} d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.46, size = 219, normalized size = 3.27 \[ -\frac {{\left (b c^{2} + a d^{2}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{4} d} - \frac {2 \, {\left (b c^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + a d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, a c d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 4 \, b c^{4} + 12 \, a c^{2} d^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} + 2 \, c {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 8 \, c^{3}\right )} c^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 48, normalized size = 0.72 \[ \frac {-2 a \,d^{2} x^{2}-b \,c^{2} x^{2}+a \,c^{2}}{\sqrt {d x +c}\, \sqrt {d x -c}\, c^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 71, normalized size = 1.06 \[ -\frac {b x}{\sqrt {d^{2} x^{2} - c^{2}} c^{2}} - \frac {2 \, a d^{2} x}{\sqrt {d^{2} x^{2} - c^{2}} c^{4}} + \frac {a}{\sqrt {d^{2} x^{2} - c^{2}} c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.87, size = 73, normalized size = 1.09 \[ \frac {2\,a\,d^2\,x^2\,\sqrt {d\,x-c}-a\,c^2\,\sqrt {d\,x-c}+b\,c^2\,x^2\,\sqrt {d\,x-c}}{c^4\,x\,\sqrt {c+d\,x}\,\left (c-d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 136.13, size = 165, normalized size = 2.46 \[ a \left (- \frac {d {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & \frac {3}{2}, \frac {5}{2}, 3 \\\frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2}, 3 & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{4}} + \frac {i d {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, 1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 1 & \\\frac {5}{4}, \frac {7}{4} & \frac {1}{2}, 1, 2, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{4}}\right ) + b \left (- \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & \frac {1}{2}, \frac {3}{2}, 2 \\\frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 2 & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{2} d} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1 & \\\frac {1}{4}, \frac {3}{4} & - \frac {1}{2}, 0, 1, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{2} d}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________