Optimal. Leaf size=196 \[ \frac {x \left (c+d x^2\right )^{3/2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{192 d^2}+\frac {c x \sqrt {c+d x^2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{128 d^2}+\frac {c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{5/2}}-\frac {b x \left (c+d x^2\right )^{5/2} (3 b c-10 a d)}{48 d^2}+\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d} \]
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Rubi [A] time = 0.12, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {416, 388, 195, 217, 206} \[ \frac {x \left (c+d x^2\right )^{3/2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{192 d^2}+\frac {c x \sqrt {c+d x^2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{128 d^2}+\frac {c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{5/2}}-\frac {b x \left (c+d x^2\right )^{5/2} (3 b c-10 a d)}{48 d^2}+\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 388
Rule 416
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d}+\frac {\int \left (c+d x^2\right )^{3/2} \left (-a (b c-8 a d)-b (3 b c-10 a d) x^2\right ) \, dx}{8 d}\\ &=-\frac {b (3 b c-10 a d) x \left (c+d x^2\right )^{5/2}}{48 d^2}+\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d}-\frac {(-b c (3 b c-10 a d)+6 a d (b c-8 a d)) \int \left (c+d x^2\right )^{3/2} \, dx}{48 d^2}\\ &=\frac {\left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{192 d^2}-\frac {b (3 b c-10 a d) x \left (c+d x^2\right )^{5/2}}{48 d^2}+\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d}+\frac {\left (c \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right )\right ) \int \sqrt {c+d x^2} \, dx}{64 d^2}\\ &=\frac {c \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \sqrt {c+d x^2}}{128 d^2}+\frac {\left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{192 d^2}-\frac {b (3 b c-10 a d) x \left (c+d x^2\right )^{5/2}}{48 d^2}+\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d}+\frac {\left (c^2 \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{128 d^2}\\ &=\frac {c \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \sqrt {c+d x^2}}{128 d^2}+\frac {\left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{192 d^2}-\frac {b (3 b c-10 a d) x \left (c+d x^2\right )^{5/2}}{48 d^2}+\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d}+\frac {\left (c^2 \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{128 d^2}\\ &=\frac {c \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \sqrt {c+d x^2}}{128 d^2}+\frac {\left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{192 d^2}-\frac {b (3 b c-10 a d) x \left (c+d x^2\right )^{5/2}}{48 d^2}+\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d}+\frac {c^2 \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 159, normalized size = 0.81 \[ \frac {3 c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )+\sqrt {d} x \sqrt {c+d x^2} \left (48 a^2 d^2 \left (5 c+2 d x^2\right )+16 a b d \left (3 c^2+14 c d x^2+8 d^2 x^4\right )+b^2 \left (-9 c^3+6 c^2 d x^2+72 c d^2 x^4+48 d^3 x^6\right )\right )}{384 d^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 344, normalized size = 1.76 \[ \left [\frac {3 \, {\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (48 \, b^{2} d^{4} x^{7} + 8 \, {\left (9 \, b^{2} c d^{3} + 16 \, a b d^{4}\right )} x^{5} + 2 \, {\left (3 \, b^{2} c^{2} d^{2} + 112 \, a b c d^{3} + 48 \, a^{2} d^{4}\right )} x^{3} - 3 \, {\left (3 \, b^{2} c^{3} d - 16 \, a b c^{2} d^{2} - 80 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{768 \, d^{3}}, -\frac {3 \, {\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (48 \, b^{2} d^{4} x^{7} + 8 \, {\left (9 \, b^{2} c d^{3} + 16 \, a b d^{4}\right )} x^{5} + 2 \, {\left (3 \, b^{2} c^{2} d^{2} + 112 \, a b c d^{3} + 48 \, a^{2} d^{4}\right )} x^{3} - 3 \, {\left (3 \, b^{2} c^{3} d - 16 \, a b c^{2} d^{2} - 80 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{384 \, d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 175, normalized size = 0.89 \[ \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, b^{2} d x^{2} + \frac {9 \, b^{2} c d^{6} + 16 \, a b d^{7}}{d^{6}}\right )} x^{2} + \frac {3 \, b^{2} c^{2} d^{5} + 112 \, a b c d^{6} + 48 \, a^{2} d^{7}}{d^{6}}\right )} x^{2} - \frac {3 \, {\left (3 \, b^{2} c^{3} d^{4} - 16 \, a b c^{2} d^{5} - 80 \, a^{2} c d^{6}\right )}}{d^{6}}\right )} \sqrt {d x^{2} + c} x - \frac {{\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{128 \, d^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 249, normalized size = 1.27 \[ \frac {3 a^{2} c^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{8 \sqrt {d}}-\frac {a b \,c^{3} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{8 d^{\frac {3}{2}}}+\frac {3 b^{2} c^{4} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {5}{2}}}+\frac {3 \sqrt {d \,x^{2}+c}\, a^{2} c x}{8}-\frac {\sqrt {d \,x^{2}+c}\, a b \,c^{2} x}{8 d}+\frac {3 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} x}{128 d^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} x^{3}}{8 d}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} x}{4}-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a b c x}{12 d}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} c^{2} x}{64 d^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a b x}{3 d}-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} c x}{16 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 227, normalized size = 1.16 \[ \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{3}}{8 \, d} + \frac {1}{4} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} x + \frac {3}{8} \, \sqrt {d x^{2} + c} a^{2} c x - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x}{16 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{2} x}{64 \, d^{2}} + \frac {3 \, \sqrt {d x^{2} + c} b^{2} c^{3} x}{128 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x}{3 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c x}{12 \, d} - \frac {\sqrt {d x^{2} + c} a b c^{2} x}{8 \, d} + \frac {3 \, b^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {5}{2}}} - \frac {a b c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {3}{2}}} + \frac {3 \, a^{2} c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {d}} \]
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 \[ \int {\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 30.55, size = 440, normalized size = 2.24 \[ \frac {a^{2} c^{\frac {3}{2}} x \sqrt {1 + \frac {d x^{2}}{c}}}{2} + \frac {a^{2} c^{\frac {3}{2}} x}{8 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a^{2} \sqrt {c} d x^{3}}{8 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a^{2} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{8 \sqrt {d}} + \frac {a^{2} d^{2} x^{5}}{4 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {a b c^{\frac {5}{2}} x}{8 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {17 a b c^{\frac {3}{2}} x^{3}}{24 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {11 a b \sqrt {c} d x^{5}}{12 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a b c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{8 d^{\frac {3}{2}}} + \frac {a b d^{2} x^{7}}{3 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {3 b^{2} c^{\frac {7}{2}} x}{128 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{\frac {5}{2}} x^{3}}{128 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {13 b^{2} c^{\frac {3}{2}} x^{5}}{64 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 b^{2} \sqrt {c} d x^{7}}{16 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 b^{2} c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {5}{2}}} + \frac {b^{2} d^{2} x^{9}}{8 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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