Optimal. Leaf size=164 \[ \frac {c^2 x \sqrt {d x-c} \sqrt {c+d x} \left (6 a d^2+5 b c^2\right )}{16 d^6}+\frac {x^3 \sqrt {d x-c} \sqrt {c+d x} \left (6 a d^2+5 b c^2\right )}{24 d^4}+\frac {c^4 \left (6 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{8 d^7}+\frac {b x^5 \sqrt {d x-c} \sqrt {c+d x}}{6 d^2} \]
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Rubi [A] time = 0.12, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {460, 100, 12, 90, 63, 217, 206} \[ \frac {x^3 \sqrt {d x-c} \sqrt {c+d x} \left (6 a d^2+5 b c^2\right )}{24 d^4}+\frac {c^2 x \sqrt {d x-c} \sqrt {c+d x} \left (6 a d^2+5 b c^2\right )}{16 d^6}+\frac {c^4 \left (6 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{8 d^7}+\frac {b x^5 \sqrt {d x-c} \sqrt {c+d x}}{6 d^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 90
Rule 100
Rule 206
Rule 217
Rule 460
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx &=\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}-\frac {1}{6} \left (-6 a-\frac {5 b c^2}{d^2}\right ) \int \frac {x^4}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx\\ &=\frac {\left (5 b c^2+6 a d^2\right ) x^3 \sqrt {-c+d x} \sqrt {c+d x}}{24 d^4}+\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}+\frac {\left (5 b c^2+6 a d^2\right ) \int \frac {3 c^2 x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{24 d^4}\\ &=\frac {\left (5 b c^2+6 a d^2\right ) x^3 \sqrt {-c+d x} \sqrt {c+d x}}{24 d^4}+\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}+\frac {\left (c^2 \left (5 b c^2+6 a d^2\right )\right ) \int \frac {x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{8 d^4}\\ &=\frac {c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^6}+\frac {\left (5 b c^2+6 a d^2\right ) x^3 \sqrt {-c+d x} \sqrt {c+d x}}{24 d^4}+\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}+\frac {\left (c^2 \left (5 b c^2+6 a d^2\right )\right ) \int \frac {c^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{16 d^6}\\ &=\frac {c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^6}+\frac {\left (5 b c^2+6 a d^2\right ) x^3 \sqrt {-c+d x} \sqrt {c+d x}}{24 d^4}+\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}+\frac {\left (c^4 \left (5 b c^2+6 a d^2\right )\right ) \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{16 d^6}\\ &=\frac {c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^6}+\frac {\left (5 b c^2+6 a d^2\right ) x^3 \sqrt {-c+d x} \sqrt {c+d x}}{24 d^4}+\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}+\frac {\left (c^4 \left (5 b c^2+6 a d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right )}{8 d^7}\\ &=\frac {c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^6}+\frac {\left (5 b c^2+6 a d^2\right ) x^3 \sqrt {-c+d x} \sqrt {c+d x}}{24 d^4}+\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}+\frac {\left (c^4 \left (5 b c^2+6 a d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 d^7}\\ &=\frac {c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^6}+\frac {\left (5 b c^2+6 a d^2\right ) x^3 \sqrt {-c+d x} \sqrt {c+d x}}{24 d^4}+\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}+\frac {c^4 \left (5 b c^2+6 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 d^7}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 148, normalized size = 0.90 \[ \frac {3 c^4 \sqrt {d^2 x^2-c^2} \left (6 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac {d x}{\sqrt {d^2 x^2-c^2}}\right )+d x \left (d^2 x^2-c^2\right ) \left (6 a d^2 \left (3 c^2+2 d^2 x^2\right )+b \left (15 c^4+10 c^2 d^2 x^2+8 d^4 x^4\right )\right )}{48 d^7 \sqrt {d x-c} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 115, normalized size = 0.70 \[ \frac {{\left (8 \, b d^{5} x^{5} + 2 \, {\left (5 \, b c^{2} d^{3} + 6 \, a d^{5}\right )} x^{3} + 3 \, {\left (5 \, b c^{4} d + 6 \, a c^{2} d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} - 3 \, {\left (5 \, b c^{6} + 6 \, a c^{4} d^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{48 \, d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 203, normalized size = 1.24 \[ \frac {{\left ({\left (2 \, {\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{6}} - \frac {5 \, b c}{d^{6}}\right )} + \frac {3 \, {\left (15 \, b c^{2} d^{36} + 2 \, a d^{38}\right )}}{d^{42}}\right )} - \frac {55 \, b c^{3} d^{36} + 18 \, a c d^{38}}{d^{42}}\right )} {\left (d x + c\right )} + \frac {85 \, b c^{4} d^{36} + 54 \, a c^{2} d^{38}}{d^{42}}\right )} {\left (d x + c\right )} - \frac {3 \, {\left (11 \, b c^{5} d^{36} + 10 \, a c^{3} d^{38}\right )}}{d^{42}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {6 \, {\left (5 \, b c^{6} + 6 \, a c^{4} d^{2}\right )} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{6}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 240, normalized size = 1.46 \[ \frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (8 \sqrt {d^{2} x^{2}-c^{2}}\, b \,d^{5} x^{5} \mathrm {csgn}\relax (d )+12 \sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{5} x^{3} \mathrm {csgn}\relax (d )+10 \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{2} d^{3} x^{3} \mathrm {csgn}\relax (d )+18 a \,c^{4} d^{2} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )+18 \sqrt {d^{2} x^{2}-c^{2}}\, a \,c^{2} d^{3} x \,\mathrm {csgn}\relax (d )+15 b \,c^{6} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )+15 \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{4} d x \,\mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{48 \sqrt {d^{2} x^{2}-c^{2}}\, d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 196, normalized size = 1.20 \[ \frac {\sqrt {d^{2} x^{2} - c^{2}} b x^{5}}{6 \, d^{2}} + \frac {5 \, \sqrt {d^{2} x^{2} - c^{2}} b c^{2} x^{3}}{24 \, d^{4}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a x^{3}}{4 \, d^{2}} + \frac {5 \, b c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{16 \, d^{7}} + \frac {3 \, a c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{8 \, d^{5}} + \frac {5 \, \sqrt {d^{2} x^{2} - c^{2}} b c^{4} x}{16 \, d^{6}} + \frac {3 \, \sqrt {d^{2} x^{2} - c^{2}} a c^{2} x}{8 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 42.66, size = 1682, normalized size = 10.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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