Optimal. Leaf size=172 \[ -\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {424, 540, 396,
223, 212} \begin {gather*} \frac {x \left (c+d x^2\right ) (b c-a d) (5 a d+2 b c)}{3 a^2 b^2 \sqrt {a+b x^2}}-\frac {d x \sqrt {a+b x^2} \left (-15 a^2 d^2+8 a b c d+4 b^2 c^2\right )}{6 a^2 b^3}+\frac {d^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 396
Rule 424
Rule 540
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {\left (c+d x^2\right ) \left (c (2 b c+a d)-d (2 b c-5 a d) x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b}\\ &=\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {a c d (2 b c-5 a d)+d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x^2}{\sqrt {a+b x^2}} \, dx}{3 a^2 b^2}\\ &=-\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (d^2 (6 b c-5 a d)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^3}\\ &=-\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (d^2 (6 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b^3}\\ &=-\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 143, normalized size = 0.83 \begin {gather*} \frac {x \left (15 a^4 d^3+4 b^4 c^3 x^2+3 a^2 b^2 d^2 x^2 \left (-8 c+d x^2\right )+6 a b^3 c^2 \left (c+d x^2\right )+2 a^3 b d^2 \left (-9 c+10 d x^2\right )\right )}{6 a^2 b^3 \left (a+b x^2\right )^{3/2}}+\frac {d^2 (-6 b c+5 a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 246, normalized size = 1.43
method | result | size |
default | \(d^{3} \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+3 c \,d^{2} \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+3 c^{2} d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )+c^{3} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )\) | \(246\) |
risch | \(\frac {d^{3} x \sqrt {b \,x^{2}+a}}{2 b^{3}}+\frac {3 d^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c}{b^{\frac {5}{2}}}-\frac {5 d^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a}{2 b^{\frac {7}{2}}}+\frac {a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, d^{3}}{12 b^{4} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c^{3}}{12 b a \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {2 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c \,d^{2}}{b^{3} \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c^{2} d}{2 b^{2} a \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, d^{3}}{12 b^{4} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c^{3}}{12 b a \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {2 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c \,d^{2}}{b^{3} \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c^{2} d}{2 b^{2} a \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c \,d^{2}}{4 b^{3} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c^{2} d}{4 b^{2} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c \,d^{2}}{4 b^{3} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c^{2} d}{4 b^{2} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {7 a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, d^{3}}{6 b^{4} \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, c^{3}}{3 b \,a^{2} \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {7 a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, d^{3}}{6 b^{4} \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, c^{3}}{3 b \,a^{2} \left (x -\frac {\sqrt {-a b}}{b}\right )}\) | \(1154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 254, normalized size = 1.48 \begin {gather*} \frac {d^{3} x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - c d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {5 \, a d^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{6 \, b} + \frac {2 \, c^{3} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{3} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {c^{2} d x}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {c^{2} d x}{\sqrt {b x^{2} + a} a b} - \frac {c d^{2} x}{\sqrt {b x^{2} + a} b^{2}} + \frac {5 \, a d^{3} x}{6 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} - \frac {5 \, a d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 486, normalized size = 2.83 \begin {gather*} \left [-\frac {3 \, {\left (6 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (6 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (6 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (3 \, a^{2} b^{3} d^{3} x^{5} + 2 \, {\left (2 \, b^{5} c^{3} + 3 \, a b^{4} c^{2} d - 12 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (2 \, a b^{4} c^{3} - 6 \, a^{3} b^{2} c d^{2} + 5 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, {\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}, -\frac {3 \, {\left (6 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (6 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (6 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, a^{2} b^{3} d^{3} x^{5} + 2 \, {\left (2 \, b^{5} c^{3} + 3 \, a b^{4} c^{2} d - 12 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (2 \, a b^{4} c^{3} - 6 \, a^{3} b^{2} c d^{2} + 5 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 158, normalized size = 0.92 \begin {gather*} \frac {{\left ({\left (\frac {3 \, d^{3} x^{2}}{b} + \frac {2 \, {\left (2 \, b^{6} c^{3} + 3 \, a b^{5} c^{2} d - 12 \, a^{2} b^{4} c d^{2} + 10 \, a^{3} b^{3} d^{3}\right )}}{a^{2} b^{5}}\right )} x^{2} + \frac {3 \, {\left (2 \, a b^{5} c^{3} - 6 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )}}{a^{2} b^{5}}\right )} x}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (6 \, b c d^{2} - 5 \, a d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, 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 = -0.01 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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