Optimal. Leaf size=161 \[ -\frac {\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {3 \left (5 b c^2+4 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{8 d^6}+\frac {3 c^2 \left (5 b c^2+4 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{4 d^7} \]
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Rubi [A]
time = 0.08, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {471, 100, 21,
92, 12, 65, 223, 212} \begin {gather*} \frac {3 c^2 \left (4 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{4 d^7}+\frac {3 x \sqrt {d x-c} \sqrt {c+d x} \left (4 a d^2+5 b c^2\right )}{8 d^6}-\frac {x^3 \left (4 a d^2+5 b c^2\right )}{4 d^4 \sqrt {d x-c} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {d x-c} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 65
Rule 92
Rule 100
Rule 212
Rule 223
Rule 471
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {1}{4} \left (-4 a-\frac {5 b c^2}{d^2}\right ) \int \frac {x^4}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\\ &=-\frac {\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (4 a+\frac {5 b c^2}{d^2}\right ) \int \frac {x^2 \left (-3 c^2-3 c d x\right )}{\sqrt {-c+d x} (c+d x)^{3/2}} \, dx}{4 c d^2}\\ &=-\frac {\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {\left (3 \left (5 b c^2+4 a d^2\right )\right ) \int \frac {x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{4 d^4}\\ &=-\frac {\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {3 \left (5 b c^2+4 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{8 d^6}+\frac {\left (3 \left (5 b c^2+4 a d^2\right )\right ) \int \frac {c^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{8 d^6}\\ &=-\frac {\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {3 \left (5 b c^2+4 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{8 d^6}+\frac {\left (3 c^2 \left (5 b c^2+4 a d^2\right )\right ) \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{8 d^6}\\ &=-\frac {\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {3 \left (5 b c^2+4 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{8 d^6}+\frac {\left (3 c^2 \left (5 b c^2+4 a d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right )}{4 d^7}\\ &=-\frac {\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {3 \left (5 b c^2+4 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{8 d^6}+\frac {\left (3 c^2 \left (5 b c^2+4 a d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{4 d^7}\\ &=-\frac {\left (5 b c^2+4 a d^2\right ) x^3}{4 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^5}{4 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {3 \left (5 b c^2+4 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{8 d^6}+\frac {3 c^2 \left (5 b c^2+4 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{4 d^7}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 137, normalized size = 0.85 \begin {gather*} \frac {4 a d^3 x \left (-3 c^2+d^2 x^2\right )+b d x \left (-15 c^4+5 c^2 d^2 x^2+2 d^4 x^4\right )+6 c^2 \left (5 b c^2+4 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {-c+d x}}\right )}{8 d^7 \sqrt {-c+d x} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.32, size = 324, normalized size = 2.01
method | result | size |
default | \(\frac {\sqrt {d x -c}\, \left (-2 \,\mathrm {csgn}\left (d \right ) b \,d^{5} x^{5} \sqrt {d^{2} x^{2}-c^{2}}-4 \,\mathrm {csgn}\left (d \right ) a \,d^{5} x^{3} \sqrt {d^{2} x^{2}-c^{2}}-5 \,\mathrm {csgn}\left (d \right ) b \,c^{2} d^{3} x^{3} \sqrt {d^{2} x^{2}-c^{2}}-12 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right )+d x \right ) \mathrm {csgn}\left (d \right )\right ) a \,c^{2} d^{4} x^{2}-15 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right )+d x \right ) \mathrm {csgn}\left (d \right )\right ) b \,c^{4} d^{2} x^{2}+12 \sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right ) d^{3} a \,c^{2} x +15 \sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right ) d b \,c^{4} x +12 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right )+d x \right ) \mathrm {csgn}\left (d \right )\right ) a \,c^{4} d^{2}+15 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right )+d x \right ) \mathrm {csgn}\left (d \right )\right ) b \,c^{6}\right ) \mathrm {csgn}\left (d \right )}{8 \left (-d x +c \right ) \sqrt {d^{2} x^{2}-c^{2}}\, d^{7} \sqrt {d x +c}}\) | \(324\) |
risch | \(-\frac {x \left (2 b \,d^{2} x^{2}+4 a \,d^{2}+7 b \,c^{2}\right ) \left (-d x +c \right ) \sqrt {d x +c}}{8 d^{6} \sqrt {d x -c}}-\frac {\left (-\frac {3 c^{2} \ln \left (\frac {d^{2} x}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) a}{2 d^{4} \sqrt {d^{2}}}-\frac {15 c^{4} \ln \left (\frac {d^{2} x}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) b}{8 d^{6} \sqrt {d^{2}}}+\frac {c^{2} \sqrt {d^{2} \left (x +\frac {c}{d}\right )^{2}-2 c d \left (x +\frac {c}{d}\right )}\, a}{2 d^{6} \left (x +\frac {c}{d}\right )}+\frac {c^{4} \sqrt {d^{2} \left (x +\frac {c}{d}\right )^{2}-2 c d \left (x +\frac {c}{d}\right )}\, b}{2 d^{8} \left (x +\frac {c}{d}\right )}+\frac {c^{2} \sqrt {d^{2} \left (x -\frac {c}{d}\right )^{2}+2 c d \left (x -\frac {c}{d}\right )}\, a}{2 d^{6} \left (x -\frac {c}{d}\right )}+\frac {c^{4} \sqrt {d^{2} \left (x -\frac {c}{d}\right )^{2}+2 c d \left (x -\frac {c}{d}\right )}\, b}{2 d^{8} \left (x -\frac {c}{d}\right )}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{\sqrt {d x -c}\, \sqrt {d x +c}}\) | \(354\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 196, normalized size = 1.22 \begin {gather*} \frac {b x^{5}}{4 \, \sqrt {d^{2} x^{2} - c^{2}} d^{2}} + \frac {5 \, b c^{2} x^{3}}{8 \, \sqrt {d^{2} x^{2} - c^{2}} d^{4}} + \frac {a x^{3}}{2 \, \sqrt {d^{2} x^{2} - c^{2}} d^{2}} - \frac {15 \, b c^{4} x}{8 \, \sqrt {d^{2} x^{2} - c^{2}} d^{6}} - \frac {3 \, a c^{2} x}{2 \, \sqrt {d^{2} x^{2} - c^{2}} d^{4}} + \frac {15 \, b c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{8 \, d^{7}} + \frac {3 \, a c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{2 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.14, size = 190, normalized size = 1.18 \begin {gather*} \frac {8 \, b c^{6} + 8 \, a c^{4} d^{2} - 8 \, {\left (b c^{4} d^{2} + a c^{2} d^{4}\right )} x^{2} + {\left (2 \, b d^{5} x^{5} + {\left (5 \, b c^{2} d^{3} + 4 \, a d^{5}\right )} x^{3} - 3 \, {\left (5 \, b c^{4} d + 4 \, a c^{2} d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} + 3 \, {\left (5 \, b c^{6} + 4 \, a c^{4} d^{2} - {\left (5 \, b c^{4} d^{2} + 4 \, a c^{2} d^{4}\right )} x^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{8 \, {\left (d^{9} x^{2} - c^{2} d^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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.
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Giac [A]
time = 0.60, size = 214, normalized size = 1.33 \begin {gather*} \frac {{\left ({\left ({\left (d x + c\right )} {\left (2 \, {\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{7}} - \frac {5 \, b c}{d^{7}}\right )} + \frac {25 \, b c^{2} d^{35} + 4 \, a d^{37}}{d^{42}}\right )} - \frac {35 \, b c^{3} d^{35} + 12 \, a c d^{37}}{d^{42}}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (7 \, b c^{4} d^{35} + 2 \, a c^{2} d^{37}\right )}}{d^{42}}\right )} \sqrt {d x + c}}{8 \, \sqrt {d x - c}} - \frac {3 \, {\left (5 \, b c^{4} + 4 \, a c^{2} d^{2}\right )} \log \left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}\right )}{8 \, d^{7}} - \frac {2 \, {\left (b c^{5} + a c^{3} d^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, c\right )} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,\left (b\,x^2+a\right )}{{\left (c+d\,x\right )}^{3/2}\,{\left (d\,x-c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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