Optimal. Leaf size=208 \[ \frac {c^2 x (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{64 d^6}+\frac {x^3 (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{48 d^4}-\frac {c^6 \left (8 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{64 d^7}+\frac {c^4 x \sqrt {d x-c} \sqrt {c+d x} \left (8 a d^2+5 b c^2\right )}{128 d^6}+\frac {b x^5 (d x-c)^{3/2} (c+d x)^{3/2}}{8 d^2} \]
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Rubi [A] time = 0.15, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {460, 100, 12, 90, 38, 63, 217, 206} \[ \frac {x^3 (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{48 d^4}+\frac {c^4 x \sqrt {d x-c} \sqrt {c+d x} \left (8 a d^2+5 b c^2\right )}{128 d^6}+\frac {c^2 x (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{64 d^6}-\frac {c^6 \left (8 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{64 d^7}+\frac {b x^5 (d x-c)^{3/2} (c+d x)^{3/2}}{8 d^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 38
Rule 63
Rule 90
Rule 100
Rule 206
Rule 217
Rule 460
Rubi steps
\begin {align*} \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx &=\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac {1}{8} \left (-8 a-\frac {5 b c^2}{d^2}\right ) \int x^4 \sqrt {-c+d x} \sqrt {c+d x} \, dx\\ &=\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}+\frac {\left (5 b c^2+8 a d^2\right ) \int 3 c^2 x^2 \sqrt {-c+d x} \sqrt {c+d x} \, dx}{48 d^4}\\ &=\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}+\frac {\left (c^2 \left (5 b c^2+8 a d^2\right )\right ) \int x^2 \sqrt {-c+d x} \sqrt {c+d x} \, dx}{16 d^4}\\ &=\frac {c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}+\frac {\left (c^2 \left (5 b c^2+8 a d^2\right )\right ) \int c^2 \sqrt {-c+d x} \sqrt {c+d x} \, dx}{64 d^6}\\ &=\frac {c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}+\frac {\left (c^4 \left (5 b c^2+8 a d^2\right )\right ) \int \sqrt {-c+d x} \sqrt {c+d x} \, dx}{64 d^6}\\ &=\frac {c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{128 d^6}+\frac {c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac {\left (c^6 \left (5 b c^2+8 a d^2\right )\right ) \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{128 d^6}\\ &=\frac {c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{128 d^6}+\frac {c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac {\left (c^6 \left (5 b c^2+8 a d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right )}{64 d^7}\\ &=\frac {c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{128 d^6}+\frac {c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac {\left (c^6 \left (5 b c^2+8 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 )}{64 d^7}\\ &=\frac {c^4 \left (5 b c^2+8 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{128 d^6}+\frac {c^2 \left (5 b c^2+8 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{64 d^6}+\frac {\left (5 b c^2+8 a d^2\right ) x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{48 d^4}+\frac {b x^5 (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^2}-\frac {c^6 \left (5 b c^2+8 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{64 d^7}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 161, normalized size = 0.77 \[ \frac {\sqrt {d x-c} \sqrt {c+d x} \left (3 \left (8 a c^5 d^2+5 b c^7\right ) \sin ^{-1}\left (\frac {d x}{c}\right )-d x \sqrt {1-\frac {d^2 x^2}{c^2}} \left (8 a d^2 \left (3 c^4+2 c^2 d^2 x^2-8 d^4 x^4\right )+b \left (15 c^6+10 c^4 d^2 x^2+8 c^2 d^4 x^4-48 d^6 x^6\right )\right )\right )}{384 d^7 \sqrt {1-\frac {d^2 x^2}{c^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 138, normalized size = 0.66 \[ \frac {{\left (48 \, b d^{7} x^{7} - 8 \, {\left (b c^{2} d^{5} - 8 \, a d^{7}\right )} x^{5} - 2 \, {\left (5 \, b c^{4} d^{3} + 8 \, a c^{2} d^{5}\right )} x^{3} - 3 \, {\left (5 \, b c^{6} d + 8 \, a c^{4} d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} + 3 \, {\left (5 \, b c^{8} + 8 \, a c^{6} d^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{384 \, d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.63, size = 558, normalized size = 2.68 \[ \frac {112 \, {\left ({\left ({\left (2 \, {\left (d x + c\right )} {\left (3 \, {\left (d x + c\right )} {\left (\frac {4 \, {\left (d x + c\right )}}{d^{4}} - \frac {21 \, c}{d^{4}}\right )} + \frac {133 \, c^{2}}{d^{4}}\right )} - \frac {295 \, c^{3}}{d^{4}}\right )} {\left (d x + c\right )} + \frac {195 \, c^{4}}{d^{4}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {90 \, c^{5} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{4}}\right )} a c + 8 \, {\left ({\left ({\left (2 \, {\left ({\left (4 \, {\left (d x + c\right )} {\left (5 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )}}{d^{6}} - \frac {43 \, c}{d^{6}}\right )} + \frac {661 \, c^{2}}{d^{6}}\right )} - \frac {4551 \, c^{3}}{d^{6}}\right )} {\left (d x + c\right )} + \frac {4781 \, c^{4}}{d^{6}}\right )} {\left (d x + c\right )} - \frac {6335 \, c^{5}}{d^{6}}\right )} {\left (d x + c\right )} + \frac {2835 \, c^{6}}{d^{6}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {1050 \, c^{7} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{6}}\right )} b c + 56 \, {\left ({\left ({\left (2 \, {\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {5 \, {\left (d x + c\right )}}{d^{5}} - \frac {31 \, c}{d^{5}}\right )} + \frac {321 \, c^{2}}{d^{5}}\right )} - \frac {451 \, c^{3}}{d^{5}}\right )} {\left (d x + c\right )} + \frac {745 \, c^{4}}{d^{5}}\right )} {\left (d x + c\right )} - \frac {405 \, c^{5}}{d^{5}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {150 \, c^{6} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{5}}\right )} a d + {\left ({\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (d x + c\right )} {\left (6 \, {\left (d x + c\right )} {\left (\frac {7 \, {\left (d x + c\right )}}{d^{7}} - \frac {57 \, c}{d^{7}}\right )} + \frac {1219 \, c^{2}}{d^{7}}\right )} - \frac {12463 \, c^{3}}{d^{7}}\right )} {\left (d x + c\right )} + \frac {64233 \, c^{4}}{d^{7}}\right )} {\left (d x + c\right )} - \frac {53963 \, c^{5}}{d^{7}}\right )} {\left (d x + c\right )} + \frac {59465 \, c^{6}}{d^{7}}\right )} {\left (d x + c\right )} - \frac {23205 \, c^{7}}{d^{7}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {7350 \, c^{8} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{7}}\right )} b d}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 298, normalized size = 1.43 \[ \frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (48 \sqrt {d^{2} x^{2}-c^{2}}\, b \,d^{7} x^{7} \mathrm {csgn}\relax (d )+64 \sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{7} x^{5} \mathrm {csgn}\relax (d )-8 \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{2} d^{5} x^{5} \mathrm {csgn}\relax (d )-16 \sqrt {d^{2} x^{2}-c^{2}}\, a \,c^{2} d^{5} x^{3} \mathrm {csgn}\relax (d )-10 \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{4} d^{3} x^{3} \mathrm {csgn}\relax (d )-24 a \,c^{6} d^{2} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )-24 \sqrt {d^{2} x^{2}-c^{2}}\, a \,c^{4} d^{3} x \,\mathrm {csgn}\relax (d )-15 b \,c^{8} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )-15 \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{6} d x \,\mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{384 \sqrt {d^{2} x^{2}-c^{2}}\, d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 246, normalized size = 1.18 \[ \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b x^{5}}{8 \, d^{2}} + \frac {5 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{2} x^{3}}{48 \, d^{4}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a x^{3}}{6 \, d^{2}} - \frac {5 \, b c^{8} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{128 \, d^{7}} - \frac {a c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{16 \, d^{5}} + \frac {5 \, \sqrt {d^{2} x^{2} - c^{2}} b c^{6} x}{128 \, d^{6}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a c^{4} x}{16 \, d^{4}} + \frac {5 \, {\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{4} x}{64 \, d^{6}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a c^{2} x}{8 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 39.15, size = 2314, normalized size = 11.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \left (a + b x^{2}\right ) \sqrt {- c + d x} \sqrt {c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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