Optimal. Leaf size=258 \[ \frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^5}+\frac {\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^5 \sqrt {d}} \]
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Rubi [A]
time = 0.30, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {478, 595, 596,
537, 223, 212, 385, 211} \begin {gather*} \frac {x \sqrt {c+d x^2} \left (32 a^2 d^2-52 a b c d+19 b^2 c^2\right )}{16 b^4}+\frac {\left (-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^5 \sqrt {d}}-\frac {\sqrt {a} (3 b c-8 a d) (b c-a d)^{3/2} \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^5}+\frac {d x^3 \sqrt {c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 223
Rule 385
Rule 478
Rule 537
Rule 595
Rule 596
Rubi steps
\begin {align*} \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx &=-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x^2 \left (c+d x^2\right )^{3/2} \left (3 c+8 d x^2\right )}{a+b x^2} \, dx}{2 b}\\ &=\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x^2 \sqrt {c+d x^2} \left (6 c (3 b c-4 a d)+6 d (7 b c-8 a d) x^2\right )}{a+b x^2} \, dx}{12 b^2}\\ &=\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x^2 \left (6 c \left (12 b^2 c^2-37 a b c d+24 a^2 d^2\right )+6 d \left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{48 b^3}\\ &=\frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\int \frac {6 a c d \left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right )-6 d \left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{96 b^4 d}\\ &=\frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\left (a (3 b c-8 a d) (b c-a d)^2\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 b^5}+\frac {\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{16 b^5}\\ &=\frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\left (a (3 b c-8 a d) (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 )}{2 b^5}+\frac {\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{16 b^5}\\ &=\frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^5}+\frac {\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^5 \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.92, size = 269, normalized size = 1.04 \begin {gather*} \frac {\frac {b x \sqrt {c+d x^2} \left (96 a^3 d^2+12 a^2 b d \left (-13 c+4 d x^2\right )+a b^2 \left (57 c^2-82 c d x^2-16 d^2 x^4\right )+b^3 x^2 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )\right )}{a+b x^2}+24 \sqrt {a} \sqrt {b c-a d} \left (3 b^2 c^2-11 a b c d+8 a^2 d^2\right ) \tan ^{-1}\left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )+\frac {3 \left (-5 b^3 c^3+60 a b^2 c^2 d-120 a^2 b c d^2+64 a^3 d^3\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{\sqrt {d}}}{48 b^5} \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. \(5358\) vs.
\(2(222)=444\).
time = 0.14, size = 5359, normalized size = 20.77
method | result | size |
risch | \(\text {Expression too large to display}\) | \(3583\) |
default | \(\text {Expression too large to display}\) | \(5359\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.73, size = 1697, normalized size = 6.58 \begin {gather*} \left [-\frac {3 \, {\left (5 \, a b^{3} c^{3} - 60 \, a^{2} b^{2} c^{2} d + 120 \, a^{3} b c d^{2} - 64 \, a^{4} d^{3} + {\left (5 \, b^{4} c^{3} - 60 \, a b^{3} c^{2} d + 120 \, a^{2} b^{2} c d^{2} - 64 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 12 \, {\left (3 \, a b^{2} c^{2} d - 11 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3} + {\left (3 \, b^{3} c^{2} d - 11 \, a b^{2} c d^{2} + 8 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 2 \, {\left (8 \, b^{4} d^{3} x^{7} + 2 \, {\left (13 \, b^{4} c d^{2} - 8 \, a b^{3} d^{3}\right )} x^{5} + {\left (33 \, b^{4} c^{2} d - 82 \, a b^{3} c d^{2} + 48 \, a^{2} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (19 \, a b^{3} c^{2} d - 52 \, a^{2} b^{2} c d^{2} + 32 \, a^{3} b d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{96 \, {\left (b^{6} d x^{2} + a b^{5} d\right )}}, -\frac {3 \, {\left (5 \, a b^{3} c^{3} - 60 \, a^{2} b^{2} c^{2} d + 120 \, a^{3} b c d^{2} - 64 \, a^{4} d^{3} + {\left (5 \, b^{4} c^{3} - 60 \, a b^{3} c^{2} d + 120 \, a^{2} b^{2} c d^{2} - 64 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 6 \, {\left (3 \, a b^{2} c^{2} d - 11 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3} + {\left (3 \, b^{3} c^{2} d - 11 \, a b^{2} c d^{2} + 8 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (8 \, b^{4} d^{3} x^{7} + 2 \, {\left (13 \, b^{4} c d^{2} - 8 \, a b^{3} d^{3}\right )} x^{5} + {\left (33 \, b^{4} c^{2} d - 82 \, a b^{3} c d^{2} + 48 \, a^{2} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (19 \, a b^{3} c^{2} d - 52 \, a^{2} b^{2} c d^{2} + 32 \, a^{3} b d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{48 \, {\left (b^{6} d x^{2} + a b^{5} d\right )}}, -\frac {24 \, {\left (3 \, a b^{2} c^{2} d - 11 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3} + {\left (3 \, b^{3} c^{2} d - 11 \, a b^{2} c d^{2} + 8 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 3 \, {\left (5 \, a b^{3} c^{3} - 60 \, a^{2} b^{2} c^{2} d + 120 \, a^{3} b c d^{2} - 64 \, a^{4} d^{3} + {\left (5 \, b^{4} c^{3} - 60 \, a b^{3} c^{2} d + 120 \, a^{2} b^{2} c d^{2} - 64 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (8 \, b^{4} d^{3} x^{7} + 2 \, {\left (13 \, b^{4} c d^{2} - 8 \, a b^{3} d^{3}\right )} x^{5} + {\left (33 \, b^{4} c^{2} d - 82 \, a b^{3} c d^{2} + 48 \, a^{2} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (19 \, a b^{3} c^{2} d - 52 \, a^{2} b^{2} c d^{2} + 32 \, a^{3} b d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{96 \, {\left (b^{6} d x^{2} + a b^{5} d\right )}}, -\frac {12 \, {\left (3 \, a b^{2} c^{2} d - 11 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3} + {\left (3 \, b^{3} c^{2} d - 11 \, a b^{2} c d^{2} + 8 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 3 \, {\left (5 \, a b^{3} c^{3} - 60 \, a^{2} b^{2} c^{2} d + 120 \, a^{3} b c d^{2} - 64 \, a^{4} d^{3} + {\left (5 \, b^{4} c^{3} - 60 \, a b^{3} c^{2} d + 120 \, a^{2} b^{2} c d^{2} - 64 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (8 \, b^{4} d^{3} x^{7} + 2 \, {\left (13 \, b^{4} c d^{2} - 8 \, a b^{3} d^{3}\right )} x^{5} + {\left (33 \, b^{4} c^{2} d - 82 \, a b^{3} c d^{2} + 48 \, a^{2} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (19 \, a b^{3} c^{2} d - 52 \, a^{2} b^{2} c d^{2} + 32 \, a^{3} b d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{48 \, {\left (b^{6} d x^{2} + a b^{5} d\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 {x^{4} \left (c + d x^{2}\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 521 vs.
\(2 (222) = 444\).
time = 0.60, size = 521, normalized size = 2.02 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (\frac {4 \, d^{2} x^{2}}{b^{2}} + \frac {13 \, b^{12} c d^{5} - 12 \, a b^{11} d^{6}}{b^{14} d^{4}}\right )} x^{2} + \frac {3 \, {\left (11 \, b^{12} c^{2} d^{4} - 36 \, a b^{11} c d^{5} + 24 \, a^{2} b^{10} d^{6}\right )}}{b^{14} d^{4}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (3 \, a b^{3} c^{3} \sqrt {d} - 14 \, a^{2} b^{2} c^{2} d^{\frac {3}{2}} + 19 \, a^{3} b c d^{\frac {5}{2}} - 8 \, a^{4} d^{\frac {7}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{5}} - \frac {{\left (5 \, b^{3} c^{3} \sqrt {d} - 60 \, a b^{2} c^{2} d^{\frac {3}{2}} + 120 \, a^{2} b c d^{\frac {5}{2}} - 64 \, a^{3} d^{\frac {7}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{32 \, b^{5} d} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{3} c^{3} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b^{2} c^{2} d^{\frac {3}{2}} + 5 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} b c d^{\frac {5}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{4} d^{\frac {7}{2}} - a b^{3} c^{4} \sqrt {d} + 2 \, a^{2} b^{2} c^{3} d^{\frac {3}{2}} - a^{3} b c^{2} d^{\frac {5}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{5}} \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 \frac {x^4\,{\left (d\,x^2+c\right )}^{5/2}}{{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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