Optimal. Leaf size=291 \[ \frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}} \]
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Rubi [A]
time = 0.37, antiderivative size = 291, 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 = {488, 595, 596,
537, 223, 212, 385, 211} \begin {gather*} \frac {a^{3/2} (b c-a d)^{5/2} \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}+\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{192 b^3}+\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{128 b^4 d}-\frac {\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 223
Rule 385
Rule 488
Rule 537
Rule 595
Rule 596
Rubi steps
\begin {align*} \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx &=\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {x^4 \sqrt {c+d x^2} \left (c (8 b c-5 a d)+d (11 b c-8 a d) x^2\right )}{a+b x^2} \, dx}{8 b}\\ &=\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {x^4 \left (c \left (48 b^2 c^2-85 a b c d+40 a^2 d^2\right )+d \left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{48 b^2}\\ &=\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}-\frac {\int \frac {x^2 \left (3 a c d \left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right )-3 d \left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{192 b^3 d}\\ &=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {-3 a c d \left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right )-3 d \left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{384 b^4 d^2}\\ &=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\left (a^2 (b c-a d)^3\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{128 b^5 d}\\ &=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\left (a^2 (b c-a d)^3\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^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{128 b^5 d}\\ &=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 266, normalized size = 0.91 \begin {gather*} \frac {\frac {b x \sqrt {c+d x^2} \left (-192 a^3 d^3+48 a^2 b d^2 \left (9 c+2 d x^2\right )-8 a b^2 d \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+b^3 \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )\right )}{d}-384 a^{3/2} (b c-a d)^{5/2} \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^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{d^{3/2}}}{384 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. \(2251\) vs.
\(2(257)=514\).
time = 0.14, size = 2252, normalized size = 7.74
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1560\) |
default | \(\text {Expression too large to display}\) | \(2252\) |
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 = 12.58, size = 1443, normalized size = 4.96 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d - 240 \, a^{2} b^{2} c^{2} d^{2} + 320 \, a^{3} b c d^{3} - 128 \, a^{4} d^{4}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 192 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\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 (48 \, b^{4} d^{4} x^{7} + 8 \, {\left (17 \, b^{4} c d^{3} - 8 \, a b^{3} d^{4}\right )} x^{5} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 104 \, a b^{3} c d^{3} + 48 \, a^{2} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{4} c^{3} d - 88 \, a b^{3} c^{2} d^{2} + 144 \, a^{2} b^{2} c d^{3} - 64 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{768 \, b^{5} d^{2}}, \frac {3 \, {\left (5 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d - 240 \, a^{2} b^{2} c^{2} d^{2} + 320 \, a^{3} b c d^{3} - 128 \, a^{4} d^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + 96 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\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 (48 \, b^{4} d^{4} x^{7} + 8 \, {\left (17 \, b^{4} c d^{3} - 8 \, a b^{3} d^{4}\right )} x^{5} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 104 \, a b^{3} c d^{3} + 48 \, a^{2} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{4} c^{3} d - 88 \, a b^{3} c^{2} d^{2} + 144 \, a^{2} b^{2} c d^{3} - 64 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{384 \, b^{5} d^{2}}, \frac {384 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\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 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d - 240 \, a^{2} b^{2} c^{2} d^{2} + 320 \, a^{3} b c d^{3} - 128 \, a^{4} d^{4}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (48 \, b^{4} d^{4} x^{7} + 8 \, {\left (17 \, b^{4} c d^{3} - 8 \, a b^{3} d^{4}\right )} x^{5} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 104 \, a b^{3} c d^{3} + 48 \, a^{2} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{4} c^{3} d - 88 \, a b^{3} c^{2} d^{2} + 144 \, a^{2} b^{2} c d^{3} - 64 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{768 \, b^{5} d^{2}}, \frac {192 \, {\left (a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3} + a^{3} d^{4}\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 \, b^{4} c^{4} + 40 \, a b^{3} c^{3} d - 240 \, a^{2} b^{2} c^{2} d^{2} + 320 \, a^{3} b c d^{3} - 128 \, a^{4} d^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (48 \, b^{4} d^{4} x^{7} + 8 \, {\left (17 \, b^{4} c d^{3} - 8 \, a b^{3} d^{4}\right )} x^{5} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 104 \, a b^{3} c d^{3} + 48 \, a^{2} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{4} c^{3} d - 88 \, a b^{3} c^{2} d^{2} + 144 \, a^{2} b^{2} c d^{3} - 64 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{384 \, b^{5} d^{2}}\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}}}{a + b x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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}}{b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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