Optimal. Leaf size=121 \[ \frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {8 x}{35 a^3 c^3 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {16 x}{35 a^4 c^4 \sqrt {a+a x} \sqrt {c-c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {40, 39}
\begin {gather*} \frac {16 x}{35 a^4 c^4 \sqrt {a x+a} \sqrt {c-c x}}+\frac {8 x}{35 a^3 c^3 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac {6 x}{35 a^2 c^2 (a x+a)^{5/2} (c-c x)^{5/2}}+\frac {x}{7 a c (a x+a)^{7/2} (c-c x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 39
Rule 40
Rubi steps
\begin {align*} \int \frac {1}{(a+a x)^{9/2} (c-c x)^{9/2}} \, dx &=\frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx}{7 a c}\\ &=\frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {24 \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx}{35 a^2 c^2}\\ &=\frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {8 x}{35 a^3 c^3 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {16 \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{35 a^3 c^3}\\ &=\frac {x}{7 a c (a+a x)^{7/2} (c-c x)^{7/2}}+\frac {6 x}{35 a^2 c^2 (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {8 x}{35 a^3 c^3 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {16 x}{35 a^4 c^4 \sqrt {a+a x} \sqrt {c-c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 54, normalized size = 0.45 \begin {gather*} \frac {x \left (-35+70 x^2-56 x^4+16 x^6\right )}{35 a^4 c^4 \sqrt {a (1+x)} \sqrt {c-c x} \left (-1+x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 188.87, size = 63, normalized size = 0.52 \begin {gather*} \frac {4 \left (I \text {meijerg}\left [\left \{\left \{\frac {9}{4},\frac {11}{4},1\right \},\left \{\frac {1}{2},\frac {9}{2},5\right \}\right \},\left \{\left \{\frac {9}{4},\frac {11}{4},4,\frac {9}{2},5\right \},\left \{0\right \}\right \},\frac {1}{x^2}\right ]+\text {meijerg}\left [\left \{\left \{-\frac {1}{2},0,\frac {1}{2},\frac {7}{4},\frac {9}{4},1\right \},\left \{\right \}\right \},\left \{\left \{\frac {7}{4},\frac {9}{4}\right \},\left \{-\frac {1}{2},0,4,0\right \}\right \},\frac {\text {exp\_polar}\left [-2 I \text {Pi}\right ]}{x^2}\right ]\right )}{105 \text {Pi}^{\frac {3}{2}} a^{\frac {9}{2}} c^{\frac {9}{2}}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(220\) vs.
\(2(97)=194\).
time = 0.14, size = 221, normalized size = 1.83
method | result | size |
gosper | \(\frac {\left (1+x \right ) \left (-1+x \right ) x \left (16 x^{6}-56 x^{4}+70 x^{2}-35\right )}{35 \left (a x +a \right )^{\frac {9}{2}} \left (-c x +c \right )^{\frac {9}{2}}}\) | \(42\) |
default | \(-\frac {1}{7 a c \left (a x +a \right )^{\frac {7}{2}} \left (-c x +c \right )^{\frac {7}{2}}}+\frac {-\frac {1}{5 a c \left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {7}{2}}}+\frac {-\frac {2}{5 a c \left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {7}{2}}}+\frac {6 \left (-\frac {5}{3 a c \sqrt {a x +a}\, \left (-c x +c \right )^{\frac {7}{2}}}+\frac {5 \left (\frac {4 \sqrt {a x +a}}{7 a c \left (-c x +c \right )^{\frac {7}{2}}}+\frac {4 \left (\frac {3 \sqrt {a x +a}}{35 a c \left (-c x +c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {a x +a}}{15 a c \left (-c x +c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {a x +a}}{15 a \,c^{2} \sqrt {-c x +c}}\right )}{7 c}\right )}{c}\right )}{3 a}\right )}{5 a}}{a}}{a}\) | \(221\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 89, normalized size = 0.74 \begin {gather*} \frac {x}{7 \, {\left (-a c x^{2} + a c\right )}^{\frac {7}{2}} a c} + \frac {6 \, x}{35 \, {\left (-a c x^{2} + a c\right )}^{\frac {5}{2}} a^{2} c^{2}} + \frac {8 \, x}{35 \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} a^{3} c^{3}} + \frac {16 \, x}{35 \, \sqrt {-a c x^{2} + a c} a^{4} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.30, size = 89, normalized size = 0.74 \begin {gather*} -\frac {{\left (16 \, x^{7} - 56 \, x^{5} + 70 \, x^{3} - 35 \, x\right )} \sqrt {a x + a} \sqrt {-c x + c}}{35 \, {\left (a^{5} c^{5} x^{8} - 4 \, a^{5} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{5} c^{5} x^{2} + a^{5} c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 437 vs.
\(2 (97) = 194\).
time = 0.18, size = 607, normalized size = 5.02 \begin {gather*} 2 \left (\frac {2 \left (\left (\left (-\frac {\frac {1}{123312537600}\cdot 7046430720 a^{3} c^{3} \left |a\right | a^{2} \sqrt {a x+a} \sqrt {a x+a}}{a^{3} c^{4} a^{4}}+\frac {\frac {1}{123312537600}\cdot 44508119040 a^{4} c^{3} \left |a\right | a^{2}}{a^{3} c^{4} a^{4}}\right ) \sqrt {a x+a} \sqrt {a x+a}-\frac {\frac {1}{123312537600}\cdot 94411161600 a^{5} c^{3} \left |a\right | a^{2}}{a^{3} c^{4} a^{4}}\right ) \sqrt {a x+a} \sqrt {a x+a}+\frac {\frac {1}{123312537600}\cdot 67436544000 a^{6} c^{3} \left |a\right | a^{2}}{a^{3} c^{4} a^{4}}\right ) \sqrt {a x+a} \sqrt {2 a^{2} c-a c \left (a x+a\right )}}{\left (2 a^{2} c-a c \left (a x+a\right )\right )^{4}}-\frac {2 \left (-175 \left (\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right )^{12}+2450 a^{2} c \left (\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right )^{10}-14280 a^{4} c^{2} \left (\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right )^{8}+43120 a^{6} c^{3} \left (\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right )^{6}-66416 a^{8} c^{4} \left (\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right )^{4}+51744 a^{10} c^{5} \left (\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right )^{2}-16384 a^{12} c^{6}\right )}{1120 a c^{3} \sqrt {-a c} \left |a\right | \left (-\left (\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right )^{2}+2 a^{2} c\right )^{7}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.48, size = 66, normalized size = 0.55 \begin {gather*} -\frac {x\,\left (16\,x^6-56\,x^4+70\,x^2-35\right )}{35\,a^4\,\sqrt {a+a\,x}\,{\left (c-c\,x\right )}^{7/2}\,\left (c-x^2\,\left (c-c\,x\right )+7\,c\,x-4\,x\,\left (c-c\,x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________