Optimal. Leaf size=272 \[ \frac {3 a (4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^3}+\frac {(4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{128 b^3}+\frac {d \left (36 b^2 c^2-20 a b c d+5 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{160 b^3}+\frac {d (14 b c-5 a d) x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{80 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b}+\frac {3 a^2 (4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{7/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {427, 542, 396,
201, 223, 212} \begin {gather*} \frac {3 a^2 (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{7/2}}+\frac {d x \left (a+b x^2\right )^{5/2} \left (5 a^2 d^2-20 a b c d+36 b^2 c^2\right )}{160 b^3}+\frac {x \left (a+b x^2\right )^{3/2} (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right )}{128 b^3}+\frac {3 a x \sqrt {a+b x^2} (4 b c-a d) \left (a^2 d^2-2 a b c d+8 b^2 c^2\right )}{256 b^3}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) (14 b c-5 a d)}{80 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 396
Rule 427
Rule 542
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3 \, dx &=\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b}+\frac {\int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (c (10 b c-a d)+d (14 b c-5 a d) x^2\right ) \, dx}{10 b}\\ &=\frac {d (14 b c-5 a d) x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{80 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b}+\frac {\int \left (a+b x^2\right )^{3/2} \left (c \left (80 b^2 c^2-22 a b c d+5 a^2 d^2\right )+3 d \left (36 b^2 c^2-20 a b c d+5 a^2 d^2\right ) x^2\right ) \, dx}{80 b^2}\\ &=\frac {d \left (36 b^2 c^2-20 a b c d+5 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{160 b^3}+\frac {d (14 b c-5 a d) x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{80 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b}+\frac {\left ((4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right )\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{32 b^3}\\ &=\frac {(4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{128 b^3}+\frac {d \left (36 b^2 c^2-20 a b c d+5 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{160 b^3}+\frac {d (14 b c-5 a d) x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{80 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b}+\frac {\left (3 a (4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right )\right ) \int \sqrt {a+b x^2} \, dx}{128 b^3}\\ &=\frac {3 a (4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^3}+\frac {(4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{128 b^3}+\frac {d \left (36 b^2 c^2-20 a b c d+5 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{160 b^3}+\frac {d (14 b c-5 a d) x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{80 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b}+\frac {\left (3 a^2 (4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^3}\\ &=\frac {3 a (4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^3}+\frac {(4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{128 b^3}+\frac {d \left (36 b^2 c^2-20 a b c d+5 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{160 b^3}+\frac {d (14 b c-5 a d) x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{80 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b}+\frac {\left (3 a^2 (4 b c-a d) \left (8 b^2 c^2-2 a b c d+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 )}{256 b^3}\\ &=\frac {3 a (4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^3}+\frac {(4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{128 b^3}+\frac {d \left (36 b^2 c^2-20 a b c d+5 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{160 b^3}+\frac {d (14 b c-5 a d) x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{80 b^2}+\frac {d x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2}{10 b}+\frac {3 a^2 (4 b c-a d) \left (8 b^2 c^2-2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 225, normalized size = 0.83 \begin {gather*} \frac {\sqrt {b} x \sqrt {a+b x^2} \left (15 a^4 d^3-10 a^3 b d^2 \left (9 c+d x^2\right )+4 a^2 b^2 d \left (60 c^2+15 c d x^2+2 d^2 x^4\right )+32 b^4 x^2 \left (10 c^3+20 c^2 d x^2+15 c d^2 x^4+4 d^3 x^6\right )+16 a b^3 \left (50 c^3+70 c^2 d x^2+45 c d^2 x^4+11 d^3 x^6\right )\right )+15 a^2 \left (-32 b^3 c^3+16 a b^2 c^2 d-6 a^2 b c d^2+a^3 d^3\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{1280 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 364, normalized size = 1.34
method | result | size |
risch | \(\frac {x \left (128 b^{4} d^{3} x^{8}+176 a \,b^{3} d^{3} x^{6}+480 b^{4} c \,d^{2} x^{6}+8 a^{2} b^{2} d^{3} x^{4}+720 a \,b^{3} c \,d^{2} x^{4}+640 b^{4} c^{2} d \,x^{4}-10 a^{3} b \,d^{3} x^{2}+60 a^{2} b^{2} c \,d^{2} x^{2}+1120 a \,b^{3} c^{2} d \,x^{2}+320 b^{4} c^{3} x^{2}+15 a^{4} d^{3}-90 a^{3} b c \,d^{2}+240 a^{2} b^{2} c^{2} d +800 a \,b^{3} c^{3}\right ) \sqrt {b \,x^{2}+a}}{1280 b^{3}}-\frac {3 a^{5} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) d^{3}}{256 b^{\frac {7}{2}}}+\frac {9 a^{4} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c \,d^{2}}{128 b^{\frac {5}{2}}}-\frac {3 a^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c^{2} d}{16 b^{\frac {3}{2}}}+\frac {3 a^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c^{3}}{8 \sqrt {b}}\) | \(292\) |
default | \(d^{3} \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )+3 c \,d^{2} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+3 c^{2} d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+c^{3} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )\) | \(364\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 364, normalized size = 1.34 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d^{3} x^{5}}{10 \, b} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c d^{2} x^{3}}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a d^{3} x^{3}}{16 \, b^{2}} + \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} c^{3} x + \frac {3}{8} \, \sqrt {b x^{2} + a} a c^{3} x + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2} d x}{2 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a c^{2} d x}{8 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a^{2} c^{2} d x}{16 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a c d^{2} x}{16 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} c d^{2} x}{64 \, b^{2}} + \frac {9 \, \sqrt {b x^{2} + a} a^{3} c d^{2} x}{128 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} d^{3} x}{32 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} d^{3} x}{128 \, b^{3}} - \frac {3 \, \sqrt {b x^{2} + a} a^{4} d^{3} x}{256 \, b^{3}} + \frac {3 \, a^{2} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} - \frac {3 \, a^{3} c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} + \frac {9 \, a^{4} c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} - \frac {3 \, a^{5} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.73, size = 502, normalized size = 1.85 \begin {gather*} \left [-\frac {15 \, {\left (32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b^{2} c^{2} d + 6 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (128 \, b^{5} d^{3} x^{9} + 16 \, {\left (30 \, b^{5} c d^{2} + 11 \, a b^{4} d^{3}\right )} x^{7} + 8 \, {\left (80 \, b^{5} c^{2} d + 90 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{5} + 10 \, {\left (32 \, b^{5} c^{3} + 112 \, a b^{4} c^{2} d + 6 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{3} + 5 \, {\left (160 \, a b^{4} c^{3} + 48 \, a^{2} b^{3} c^{2} d - 18 \, a^{3} b^{2} c d^{2} + 3 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{2560 \, b^{4}}, -\frac {15 \, {\left (32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b^{2} c^{2} d + 6 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (128 \, b^{5} d^{3} x^{9} + 16 \, {\left (30 \, b^{5} c d^{2} + 11 \, a b^{4} d^{3}\right )} x^{7} + 8 \, {\left (80 \, b^{5} c^{2} d + 90 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{5} + 10 \, {\left (32 \, b^{5} c^{3} + 112 \, a b^{4} c^{2} d + 6 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{3} + 5 \, {\left (160 \, a b^{4} c^{3} + 48 \, a^{2} b^{3} c^{2} d - 18 \, a^{3} b^{2} c d^{2} + 3 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{1280 \, b^{4}}\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. 665 vs.
\(2 (269) = 538\).
time = 172.36, size = 665, normalized size = 2.44 \begin {gather*} \frac {3 a^{\frac {9}{2}} d^{3} x}{256 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {9 a^{\frac {7}{2}} c d^{2} x}{128 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {7}{2}} d^{3} x^{3}}{256 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{\frac {5}{2}} c^{2} d x}{16 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {5}{2}} c d^{2} x^{3}}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{\frac {5}{2}} d^{3} x^{5}}{640 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {3}{2}} c^{3} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} c^{3} x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} c^{2} d x^{3}}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {39 a^{\frac {3}{2}} c d^{2} x^{5}}{64 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 a^{\frac {3}{2}} d^{3} x^{7}}{160 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 \sqrt {a} b c^{3} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {11 \sqrt {a} b c^{2} d x^{5}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 \sqrt {a} b c d^{2} x^{7}}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {19 \sqrt {a} b d^{3} x^{9}}{80 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{5} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {7}{2}}} + \frac {9 a^{4} c d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {5}{2}}} - \frac {3 a^{3} c^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + \frac {3 a^{2} c^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 \sqrt {b}} + \frac {b^{2} c^{3} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{2} c^{2} d x^{7}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 b^{2} c d^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{2} d^{3} x^{11}}{10 \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 = 0.97, size = 260, normalized size = 0.96 \begin {gather*} \frac {1}{1280} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, b d^{3} x^{2} + \frac {30 \, b^{9} c d^{2} + 11 \, a b^{8} d^{3}}{b^{8}}\right )} x^{2} + \frac {80 \, b^{9} c^{2} d + 90 \, a b^{8} c d^{2} + a^{2} b^{7} d^{3}}{b^{8}}\right )} x^{2} + \frac {5 \, {\left (32 \, b^{9} c^{3} + 112 \, a b^{8} c^{2} d + 6 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )}}{b^{8}}\right )} x^{2} + \frac {5 \, {\left (160 \, a b^{8} c^{3} + 48 \, a^{2} b^{7} c^{2} d - 18 \, a^{3} b^{6} c d^{2} + 3 \, a^{4} b^{5} d^{3}\right )}}{b^{8}}\right )} \sqrt {b x^{2} + a} x - \frac {3 \, {\left (32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b^{2} c^{2} d + 6 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {7}{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 = -0.00 \begin {gather*} \int {\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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