Optimal. Leaf size=196 \[ \frac {a \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{128 b^2}+\frac {\left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^2}+\frac {d (10 b c-3 a d) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}+\frac {a^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \]
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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 (a+b x^2\right )^{3/2} \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right )}{192 b^2}+\frac {a x \sqrt {a+b x^2} \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right )}{128 b^2}+\frac {a^2 \left (3 a^2 d^2-16 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}+\frac {d x \left (a+b x^2\right )^{5/2} (10 b c-3 a d)}{48 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b} \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 )^{3/2} \left (c+d x^2\right )^2 \, dx &=\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}+\frac {\int \left (a+b x^2\right )^{3/2} \left (c (8 b c-a d)+d (10 b c-3 a d) x^2\right ) \, dx}{8 b}\\ &=\frac {d (10 b c-3 a d) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}-\frac {(a d (10 b c-3 a d)-6 b c (8 b c-a d)) \int \left (a+b x^2\right )^{3/2} \, dx}{48 b^2}\\ &=\frac {\left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^2}+\frac {d (10 b c-3 a d) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}+\frac {\left (a \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \int \sqrt {a+b x^2} \, dx}{64 b^2}\\ &=\frac {a \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{128 b^2}+\frac {\left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^2}+\frac {d (10 b c-3 a d) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}+\frac {\left (a^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b^2}\\ &=\frac {a \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{128 b^2}+\frac {\left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^2}+\frac {d (10 b c-3 a d) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}+\frac {\left (a^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b^2}\\ &=\frac {a \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{128 b^2}+\frac {\left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^2}+\frac {d (10 b c-3 a d) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}+\frac {a^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 158, normalized size = 0.81 \begin {gather*} \frac {\sqrt {b} x \sqrt {a+b x^2} \left (-9 a^3 d^2+6 a^2 b d \left (8 c+d x^2\right )+16 b^3 x^2 \left (6 c^2+8 c d x^2+3 d^2 x^4\right )+8 a b^2 \left (30 c^2+28 c d x^2+9 d^2 x^4\right )\right )-3 a^2 \left (48 b^2 c^2-16 a b c d+3 a^2 d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{384 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 235, normalized size = 1.20
method | result | size |
risch | \(-\frac {x \left (-48 b^{3} d^{2} x^{6}-72 a \,b^{2} d^{2} x^{4}-128 b^{3} c d \,x^{4}-6 a^{2} b \,d^{2} x^{2}-224 a \,b^{2} c d \,x^{2}-96 b^{3} c^{2} x^{2}+9 a^{3} d^{2}-48 a^{2} b c d -240 a \,b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}}{384 b^{2}}+\frac {3 a^{4} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) d^{2}}{128 b^{\frac {5}{2}}}-\frac {a^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c d}{8 b^{\frac {3}{2}}}+\frac {3 a^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c^{2}}{8 \sqrt {b}}\) | \(190\) |
default | \(d^{2} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+2 c d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+c^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )\) | \(235\) |
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 (b x^{2} + a\right )}^{\frac {5}{2}} d^{2} x^{3}}{8 \, b} + \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} c^{2} x + \frac {3}{8} \, \sqrt {b x^{2} + a} a c^{2} x + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c d x}{3 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a c d x}{12 \, b} - \frac {\sqrt {b x^{2} + a} a^{2} c d x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a d^{2} x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{2} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{3} d^{2} x}{128 \, b^{2}} + \frac {3 \, a^{2} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} - \frac {a^{3} c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} + \frac {3 \, a^{4} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 344, normalized size = 1.76 \begin {gather*} \left [\frac {3 \, {\left (48 \, a^{2} b^{2} c^{2} - 16 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (48 \, b^{4} d^{2} x^{7} + 8 \, {\left (16 \, b^{4} c d + 9 \, a b^{3} d^{2}\right )} x^{5} + 2 \, {\left (48 \, b^{4} c^{2} + 112 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{3} + 3 \, {\left (80 \, a b^{3} c^{2} + 16 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{3}}, -\frac {3 \, {\left (48 \, a^{2} b^{2} c^{2} - 16 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (48 \, b^{4} d^{2} x^{7} + 8 \, {\left (16 \, b^{4} c d + 9 \, a b^{3} d^{2}\right )} x^{5} + 2 \, {\left (48 \, b^{4} c^{2} + 112 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{3} + 3 \, {\left (80 \, a b^{3} c^{2} + 16 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{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 = 35.72, size = 440, normalized size = 2.24 \begin {gather*} - \frac {3 a^{\frac {7}{2}} d^{2} x}{128 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {5}{2}} c d x}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{\frac {5}{2}} d^{2} x^{3}}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {3}{2}} c^{2} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} c^{2} x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} c d x^{3}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {13 a^{\frac {3}{2}} d^{2} x^{5}}{64 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 \sqrt {a} b c^{2} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {11 \sqrt {a} b c d x^{5}}{12 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 \sqrt {a} b d^{2} x^{7}}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{4} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {5}{2}}} - \frac {a^{3} c d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} + \frac {3 a^{2} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 \sqrt {b}} + \frac {b^{2} c^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{2} c d x^{7}}{3 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{2} d^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 175, normalized size = 0.89 \begin {gather*} \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, b d^{2} x^{2} + \frac {16 \, b^{7} c d + 9 \, a b^{6} d^{2}}{b^{6}}\right )} x^{2} + \frac {48 \, b^{7} c^{2} + 112 \, a b^{6} c d + 3 \, a^{2} b^{5} d^{2}}{b^{6}}\right )} x^{2} + \frac {3 \, {\left (80 \, a b^{6} c^{2} + 16 \, a^{2} b^{5} c d - 3 \, a^{3} b^{4} d^{2}\right )}}{b^{6}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (48 \, a^{2} b^{2} c^{2} - 16 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\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 )}^{3/2}\,{\left (d\,x^2+c\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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