Optimal. Leaf size=196 \[ \frac {c \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \sqrt {c+d x^2}}{128 d^2}+\frac {\left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) x \left (c+d x^2\right )^{3/2}}{192 d^2}-\frac {b (3 b c-10 a d) x \left (c+d x^2\right )^{5/2}}{48 d^2}+\frac {b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d}+\frac {c^2 \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{5/2}} \]
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Rubi [A]
time = 0.08, 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 = {427, 396, 201,
223, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 396
Rule 427
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 ) \text {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 = 0.23, size = 158, normalized size = 0.81 \begin {gather*} \frac {\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 )-3 c^2 \left (3 b^2 c^2-16 a b c d+48 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{384 d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 235, normalized size = 1.20
method | result | size |
risch | \(\frac {x \left (48 b^{2} x^{6} d^{3}+128 a b \,d^{3} x^{4}+72 b^{2} c \,d^{2} x^{4}+96 a^{2} d^{3} x^{2}+224 a b c \,d^{2} x^{2}+6 b^{2} c^{2} d \,x^{2}+240 a^{2} c \,d^{2}+48 a b \,c^{2} d -9 b^{2} c^{3}\right ) \sqrt {d \,x^{2}+c}}{384 d^{2}}+\frac {3 c^{2} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a^{2}}{8 \sqrt {d}}-\frac {c^{3} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a b}{8 d^{\frac {3}{2}}}+\frac {3 c^{4} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) b^{2}}{128 d^{\frac {5}{2}}}\) | \(190\) |
default | \(b^{2} \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{8 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )}{8 d}\right )+2 a b \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )+a^{2} \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )\) | \(235\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 227, normalized size = 1.16 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.94, size = 344, normalized size = 1.76 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 440 vs.
\(2 (190) = 380\).
time = 35.65, size = 440, normalized size = 2.24 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.59, size = 175, normalized size = 0.89 \begin {gather*} \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}}} \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 {\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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