Optimal. Leaf size=385 \[ -\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{231 d x^7}-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {1-c^2 x^2}}+\frac {16 b c^{11} d \log (x) \sqrt {d-c^2 d x^2}}{1155 \sqrt {1-c^2 x^2}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {1-c^2 x^2}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {271, 264, 4691, 12, 1799, 1620} \[ -\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{231 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{33 d x^9}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {1-c^2 x^2}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {1-c^2 x^2}}+\frac {16 b c^{11} d \log (x) \sqrt {d-c^2 d x^2}}{1155 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 264
Rule 271
Rule 1620
Rule 1799
Rule 4691
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^{12}} \, dx &=-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (-105-70 c^2 x^2-40 c^4 x^4-16 c^6 x^6\right )}{1155 x^{11}} \, dx}{\sqrt {1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{x^{12}} \, dx\\ &=-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (-105-70 c^2 x^2-40 c^4 x^4-16 c^6 x^6\right )}{x^{11}} \, dx}{1155 \sqrt {1-c^2 x^2}}+\frac {1}{11} \left (6 c^2 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{x^{10}} \, dx\\ &=-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{33 d x^9}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-c^2 x\right )^2 \left (-105-70 c^2 x-40 c^4 x^2-16 c^6 x^3\right )}{x^6} \, dx,x,x^2\right )}{2310 \sqrt {1-c^2 x^2}}+\frac {1}{33} \left (8 c^4 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{x^8} \, dx\\ &=-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{33 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{231 d x^7}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {105}{x^6}+\frac {140 c^2}{x^5}-\frac {5 c^4}{x^4}-\frac {6 c^6}{x^3}-\frac {8 c^8}{x^2}-\frac {16 c^{10}}{x}\right ) \, dx,x,x^2\right )}{2310 \sqrt {1-c^2 x^2}}+\frac {1}{231} \left (16 c^6 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{x^6} \, dx\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {1-c^2 x^2}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {1-c^2 x^2}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {1-c^2 x^2}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {1-c^2 x^2}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{33 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{231 d x^7}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{1155 d x^5}+\frac {16 b c^{11} d \sqrt {d-c^2 d x^2} \log (x)}{1155 \sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 221, normalized size = 0.57 \[ \frac {16 b c^{11} d \log (x) \sqrt {d-c^2 d x^2}}{1155 \sqrt {1-c^2 x^2}}-\frac {d \sqrt {d-c^2 d x^2} \left (630 a \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right ) \left (c^2 x^2-1\right )^3+630 b \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right ) \left (c^2 x^2-1\right )^3 \sin ^{-1}(c x)-b c x \sqrt {1-c^2 x^2} \left (29524 c^{10} x^{10}+2520 c^8 x^8+945 c^6 x^6+525 c^4 x^4-11025 c^2 x^2+6615\right )\right )}{727650 x^{11} \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 743, normalized size = 1.93 \[ \left [\frac {48 \, {\left (b c^{13} d x^{13} - b c^{11} d x^{11}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d + {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}, \frac {96 \, {\left (b c^{13} d x^{13} - b c^{11} d x^{11}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} + 1\right )} \sqrt {-d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) + {\left (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d + {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.87, size = 5881, normalized size = 15.28 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 269, normalized size = 0.70 \[ \frac {1}{6930} \, {\left (96 \, c^{10} d^{\frac {3}{2}} \log \relax (x) - \frac {24 \, c^{8} d^{\frac {3}{2}} x^{8} + 9 \, c^{6} d^{\frac {3}{2}} x^{6} + 5 \, c^{4} d^{\frac {3}{2}} x^{4} - 105 \, c^{2} d^{\frac {3}{2}} x^{2} + 63 \, d^{\frac {3}{2}}}{x^{10}}\right )} b c - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} b \arcsin \left (c x\right ) - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{12}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________