Optimal. Leaf size=144 \[ -\frac {a (b c-a d)^2 \sqrt {c+d x^2}}{b^4}-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}+\frac {a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{9/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 81, 52, 65,
214} \begin {gather*} \frac {a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{9/2}}-\frac {a \sqrt {c+d x^2} (b c-a d)^2}{b^4}-\frac {a \left (c+d x^2\right )^{3/2} (b c-a d)}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x (c+d x)^{5/2}}{a+b x} \, dx,x,x^2\right )\\ &=\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {a \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{a+b x} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {(a (b c-a d)) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{a+b x} \, dx,x,x^2\right )}{2 b^2}\\ &=-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {\left (a (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^2\right )}{2 b^3}\\ &=-\frac {a (b c-a d)^2 \sqrt {c+d x^2}}{b^4}-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {\left (a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac {a (b c-a d)^2 \sqrt {c+d x^2}}{b^4}-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}-\frac {\left (a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{b^4 d}\\ &=-\frac {a (b c-a d)^2 \sqrt {c+d x^2}}{b^4}-\frac {a (b c-a d) \left (c+d x^2\right )^{3/2}}{3 b^3}-\frac {a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac {\left (c+d x^2\right )^{7/2}}{7 b d}+\frac {a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 140, normalized size = 0.97 \begin {gather*} \frac {\sqrt {c+d x^2} \left (-105 a^3 d^3+15 b^3 \left (c+d x^2\right )^3+35 a^2 b d^2 \left (7 c+d x^2\right )-7 a b^2 d \left (23 c^2+11 c d x^2+3 d^2 x^4\right )\right )}{105 b^4 d}+\frac {a (-b c+a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2086\) vs.
\(2(120)=240\).
time = 0.11, size = 2087, normalized size = 14.49
method | result | size |
risch | \(-\frac {\left (-15 d^{3} b^{3} x^{6}+21 a \,b^{2} d^{3} x^{4}-45 b^{3} c \,d^{2} x^{4}-35 a^{2} b \,d^{3} x^{2}+77 a \,b^{2} c \,d^{2} x^{2}-45 b^{3} c^{2} d \,x^{2}+105 a^{3} d^{3}-245 a^{2} b c \,d^{2}+161 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) \sqrt {d \,x^{2}+c}}{105 d \,b^{4}}-\frac {a^{4} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) d^{3}}{2 b^{5} \sqrt {-\frac {a d -b c}{b}}}+\frac {3 a^{3} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) c \,d^{2}}{2 b^{4} \sqrt {-\frac {a d -b c}{b}}}-\frac {3 a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) c^{2} d}{2 b^{3} \sqrt {-\frac {a d -b c}{b}}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) c^{3}}{2 b^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {a^{4} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) d^{3}}{2 b^{5} \sqrt {-\frac {a d -b c}{b}}}+\frac {3 a^{3} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) c \,d^{2}}{2 b^{4} \sqrt {-\frac {a d -b c}{b}}}-\frac {3 a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) c^{2} d}{2 b^{3} \sqrt {-\frac {a d -b c}{b}}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) c^{3}}{2 b^{2} \sqrt {-\frac {a d -b c}{b}}}\) | \(1368\) |
default | \(\text {Expression too large to display}\) | \(2087\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.64, size = 527, normalized size = 3.66 \begin {gather*} \left [\frac {105 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (15 \, b^{3} d^{3} x^{6} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \, {\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{4} + {\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{420 \, b^{4} d}, \frac {105 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (15 \, b^{3} d^{3} x^{6} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \, {\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{4} + {\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{210 \, b^{4} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 35.01, size = 144, normalized size = 1.00 \begin {gather*} - \frac {a \left (c + d x^{2}\right )^{\frac {5}{2}}}{5 b^{2}} + \frac {a \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{5} \sqrt {\frac {a d - b c}{b}}} + \frac {\left (c + d x^{2}\right )^{\frac {7}{2}}}{7 b d} + \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (a^{2} d - a b c\right )}{3 b^{3}} + \frac {\sqrt {c + d x^{2}} \left (- a^{3} d^{2} + 2 a^{2} b c d - a b^{2} c^{2}\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 228, normalized size = 1.58 \begin {gather*} -\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{4}} + \frac {15 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{6} d^{6} - 21 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b^{5} d^{7} - 35 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b^{5} c d^{7} - 105 \, \sqrt {d x^{2} + c} a b^{5} c^{2} d^{7} + 35 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} b^{4} d^{8} + 210 \, \sqrt {d x^{2} + c} a^{2} b^{4} c d^{8} - 105 \, \sqrt {d x^{2} + c} a^{3} b^{3} d^{9}}{105 \, b^{7} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 251, normalized size = 1.74 \begin {gather*} \frac {{\left (d\,x^2+c\right )}^{7/2}}{7\,b\,d}-{\left (d\,x^2+c\right )}^{5/2}\,\left (\frac {c}{5\,b\,d}+\frac {a\,d^2-b\,c\,d}{5\,b^2\,d^2}\right )+\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^2+c}\,{\left (a\,d-b\,c\right )}^{5/2}}{a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{9/2}}+\frac {{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d^2-b\,c\,d\right )\,\left (\frac {c}{b\,d}+\frac {a\,d^2-b\,c\,d}{b^2\,d^2}\right )}{3\,b\,d}-\frac {\sqrt {d\,x^2+c}\,{\left (a\,d^2-b\,c\,d\right )}^2\,\left (\frac {c}{b\,d}+\frac {a\,d^2-b\,c\,d}{b^2\,d^2}\right )}{b^2\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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