Integrand size = 24, antiderivative size = 210 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {\left (b^2 c^2-10 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^3 d}+\frac {(7 b c-6 a d) x^3 \sqrt {c+d x^2}}{24 b^2}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}+\frac {a^{3/2} (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^4}-\frac {(b c-2 a d) \left (b^2 c^2+8 a b c d-8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^4 d^{3/2}} \]
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Time = 0.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {488, 596, 537, 223, 212, 385, 211} \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {a^{3/2} (b c-a d)^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^4}-\frac {(b c-2 a d) \left (-8 a^2 d^2+8 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^4 d^{3/2}}+\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2-10 a b c d+b^2 c^2\right )}{16 b^3 d}+\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{24 b^2}+\frac {d x^5 \sqrt {c+d x^2}}{6 b} \]
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Rule 211
Rule 212
Rule 223
Rule 385
Rule 488
Rule 537
Rule 596
Rubi steps \begin{align*} \text {integral}& = \frac {d x^5 \sqrt {c+d x^2}}{6 b}+\frac {\int \frac {x^4 \left (c (6 b c-5 a d)+d (7 b c-6 a d) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 b} \\ & = \frac {(7 b c-6 a d) x^3 \sqrt {c+d x^2}}{24 b^2}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}-\frac {\int \frac {x^2 \left (3 a c d (7 b c-6 a d)-3 d \left (b^2 c^2-10 a b c d+8 a^2 d^2\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{24 b^2 d} \\ & = \frac {\left (b^2 c^2-10 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^3 d}+\frac {(7 b c-6 a d) x^3 \sqrt {c+d x^2}}{24 b^2}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}+\frac {\int \frac {-3 a c d \left (b^2 c^2-10 a b c d+8 a^2 d^2\right )-3 d (b c-2 a d) \left (b^2 c^2+8 a b c d-8 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{48 b^3 d^2} \\ & = \frac {\left (b^2 c^2-10 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^3 d}+\frac {(7 b c-6 a d) x^3 \sqrt {c+d x^2}}{24 b^2}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}+\frac {\left (a^2 (b c-a d)^2\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^4}-\frac {\left ((b c-2 a d) \left (b^2 c^2+8 a b c d-8 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{16 b^4 d} \\ & = \frac {\left (b^2 c^2-10 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^3 d}+\frac {(7 b c-6 a d) x^3 \sqrt {c+d x^2}}{24 b^2}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}+\frac {\left (a^2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^4}-\frac {\left ((b c-2 a d) \left (b^2 c^2+8 a b c d-8 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 )}{16 b^4 d} \\ & = \frac {\left (b^2 c^2-10 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^3 d}+\frac {(7 b c-6 a d) x^3 \sqrt {c+d x^2}}{24 b^2}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}+\frac {a^{3/2} (b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^4}-\frac {(b c-2 a d) \left (b^2 c^2+8 a b c d-8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^4 d^{3/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(457\) vs. \(2(210)=420\).
Time = 1.99 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.18 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {b \sqrt {d} x \sqrt {c+d x^2} \left (24 a^2 d^2-6 a b d \left (5 c+2 d x^2\right )+b^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )\right )+48 \sqrt {a} \sqrt {d} (-b c+a d) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (-b c+a d-\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )+48 \sqrt {a} \sqrt {d} (-b c+a d) \left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )+6 \left (b^3 c^3+6 a b^2 c^2 d-24 a^2 b c d^2+16 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )}{48 b^4 d^{3/2}} \]
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Time = 3.06 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(-\frac {-\frac {b \sqrt {d \,x^{2}+c}\, \left (8 b^{2} d^{2} x^{4}-12 x^{2} a b \,d^{2}+14 x^{2} b^{2} c d +24 a^{2} d^{2}-30 a b c d +3 b^{2} c^{2}\right ) x}{24 d}+\frac {\left (16 a^{3} d^{3}-24 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{8 d^{\frac {3}{2}}}-\frac {2 \left (a d -b c \right )^{2} a^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{4}}\) | \(189\) |
risch | \(\frac {x \left (8 b^{2} d^{2} x^{4}-12 x^{2} a b \,d^{2}+14 x^{2} b^{2} c d +24 a^{2} d^{2}-30 a b c d +3 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}}{48 d \,b^{3}}-\frac {\frac {\left (16 a^{3} d^{3}-24 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}-\frac {8 a^{2} d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}+\frac {8 a^{2} d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{16 b^{3} d}\) | \(502\) |
default | \(\text {Expression too large to display}\) | \(1389\) |
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Time = 1.36 (sec) , antiderivative size = 1119, normalized size of antiderivative = 5.33 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\int \frac {x^{4} \left (c + d x^{2}\right )^{\frac {3}{2}}}{a + b x^{2}}\, dx \]
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\[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}}{b x^{2} + a} \,d x } \]
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Exception generated. \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\int \frac {x^4\,{\left (d\,x^2+c\right )}^{3/2}}{b\,x^2+a} \,d x \]
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