Optimal. Leaf size=136 \[ -\frac {\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac {8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac {16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac {64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac {128 c^4 \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2041, 2039}
\begin {gather*} -\frac {128 c^4 \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}+\frac {64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac {16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac {8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac {\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^{13}} \, dx &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac {(8 c) \int \frac {\sqrt {b x^2+c x^4}}{x^{11}} \, dx}{11 b}\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac {8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac {\left (16 c^2\right ) \int \frac {\sqrt {b x^2+c x^4}}{x^9} \, dx}{33 b^2}\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac {8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac {16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac {\left (64 c^3\right ) \int \frac {\sqrt {b x^2+c x^4}}{x^7} \, dx}{231 b^3}\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac {8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac {16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac {64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}+\frac {\left (128 c^4\right ) \int \frac {\sqrt {b x^2+c x^4}}{x^5} \, dx}{1155 b^4}\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac {8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac {16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac {64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac {128 c^4 \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 79, normalized size = 0.58 \begin {gather*} \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (-315 b^5-35 b^4 c x^2+40 b^3 c^2 x^4-48 b^2 c^3 x^6+64 b c^4 x^8-128 c^5 x^{10}\right )}{3465 b^5 x^{12}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 72, normalized size = 0.53
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (128 c^{4} x^{8}-192 b \,c^{3} x^{6}+240 b^{2} c^{2} x^{4}-280 b^{3} c \,x^{2}+315 b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3465 x^{12} b^{5}}\) | \(72\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (128 c^{4} x^{8}-192 b \,c^{3} x^{6}+240 b^{2} c^{2} x^{4}-280 b^{3} c \,x^{2}+315 b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3465 x^{12} b^{5}}\) | \(72\) |
trager | \(-\frac {\left (128 c^{5} x^{10}-64 b \,c^{4} x^{8}+48 b^{2} c^{3} x^{6}-40 b^{3} c^{2} x^{4}+35 b^{4} c \,x^{2}+315 b^{5}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3465 x^{12} b^{5}}\) | \(76\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (128 c^{5} x^{10}-64 b \,c^{4} x^{8}+48 b^{2} c^{3} x^{6}-40 b^{3} c^{2} x^{4}+35 b^{4} c \,x^{2}+315 b^{5}\right )}{3465 x^{12} b^{5}}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 137, normalized size = 1.01 \begin {gather*} -\frac {128 \, \sqrt {c x^{4} + b x^{2}} c^{5}}{3465 \, b^{5} x^{2}} + \frac {64 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{3465 \, b^{4} x^{4}} - \frac {16 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{1155 \, b^{3} x^{6}} + \frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{693 \, b^{2} x^{8}} - \frac {\sqrt {c x^{4} + b x^{2}} c}{99 \, b x^{10}} - \frac {\sqrt {c x^{4} + b x^{2}}}{11 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 75, normalized size = 0.55 \begin {gather*} -\frac {{\left (128 \, c^{5} x^{10} - 64 \, b c^{4} x^{8} + 48 \, b^{2} c^{3} x^{6} - 40 \, b^{3} c^{2} x^{4} + 35 \, b^{4} c x^{2} + 315 \, b^{5}\right )} \sqrt {c x^{4} + b x^{2}}}{3465 \, b^{5} x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{13}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 5.22, size = 206, normalized size = 1.51 \begin {gather*} \frac {256 \, {\left (1386 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 924 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} b c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 330 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} b^{2} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 165 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} b^{3} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 55 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{4} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{5} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + b^{6} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right )\right )}}{3465 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.62, size = 137, normalized size = 1.01 \begin {gather*} \frac {8\,c^2\,\sqrt {c\,x^4+b\,x^2}}{693\,b^2\,x^8}-\frac {c\,\sqrt {c\,x^4+b\,x^2}}{99\,b\,x^{10}}-\frac {\sqrt {c\,x^4+b\,x^2}}{11\,x^{12}}-\frac {16\,c^3\,\sqrt {c\,x^4+b\,x^2}}{1155\,b^3\,x^6}+\frac {64\,c^4\,\sqrt {c\,x^4+b\,x^2}}{3465\,b^4\,x^4}-\frac {128\,c^5\,\sqrt {c\,x^4+b\,x^2}}{3465\,b^5\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________