Optimal. Leaf size=195 \[ \frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^4}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4} \]
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Rubi [A]
time = 0.16, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {478, 542, 537,
223, 212, 385, 211} \begin {gather*} \frac {\sqrt {d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4}+\frac {(b c-6 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 \sqrt {a} b^4}+\frac {d x \sqrt {c+d x^2} (11 b c-12 a d)}{8 b^3}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 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 542
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx &=-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {\left (c+d x^2\right )^{3/2} \left (c+6 d x^2\right )}{a+b x^2} \, dx}{2 b}\\ &=\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {\sqrt {c+d x^2} \left (2 c (2 b c-3 a d)+2 d (11 b c-12 a d) x^2\right )}{a+b x^2} \, dx}{8 b^2}\\ &=\frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {2 c \left (4 b^2 c^2-17 a b c d+12 a^2 d^2\right )+2 d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{16 b^3}\\ &=\frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-6 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^4}+\frac {\left (d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 b^4}\\ &=\frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-6 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^4}+\frac {\left (d \left (15 b^2 c^2-40 a b c d+24 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 )}{8 b^4}\\ &=\frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {(b c-6 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 \sqrt {a} b^4}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.81, size = 221, normalized size = 1.13 \begin {gather*} -\frac {\frac {b x \sqrt {c+d x^2} \left (12 a^2 d^2+a b d \left (-17 c+6 d x^2\right )+b^2 \left (4 c^2-9 c d x^2-2 d^2 x^4\right )\right )}{a+b x^2}+\frac {4 \sqrt {b c-a d} \left (b^2 c^2-7 a b c d+6 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 )}{\sqrt {a}}+\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{8 b^4} \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. \(5283\) vs.
\(2(163)=326\).
time = 0.14, size = 5284, normalized size = 27.10
method | result | size |
risch | \(\text {Expression too large to display}\) | \(3498\) |
default | \(\text {Expression too large to display}\) | \(5284\) |
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 = 3.61, size = 1379, normalized size = 7.07 \begin {gather*} \left [\frac {{\left (15 \, a b^{2} c^{2} - 40 \, a^{2} b c d + 24 \, a^{3} d^{2} + {\left (15 \, b^{3} c^{2} - 40 \, a b^{2} c d + 24 \, a^{2} b d^{2}\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 (a b^{2} c^{2} - 7 \, a^{2} b c d + 6 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 7 \, a b^{2} c d + 6 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \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 (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left (2 \, b^{3} d^{2} x^{5} + 3 \, {\left (3 \, b^{3} c d - 2 \, a b^{2} d^{2}\right )} x^{3} - {\left (4 \, b^{3} c^{2} - 17 \, a b^{2} c d + 12 \, a^{2} b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, -\frac {{\left (15 \, a b^{2} c^{2} - 40 \, a^{2} b c d + 24 \, a^{3} d^{2} + {\left (15 \, b^{3} c^{2} - 40 \, a b^{2} c d + 24 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (a b^{2} c^{2} - 7 \, a^{2} b c d + 6 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 7 \, a b^{2} c d + 6 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \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 (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b^{3} d^{2} x^{5} + 3 \, {\left (3 \, b^{3} c d - 2 \, a b^{2} d^{2}\right )} x^{3} - {\left (4 \, b^{3} c^{2} - 17 \, a b^{2} c d + 12 \, a^{2} b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, \frac {4 \, {\left (a b^{2} c^{2} - 7 \, a^{2} b c d + 6 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 7 \, a b^{2} c d + 6 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + {\left (15 \, a b^{2} c^{2} - 40 \, a^{2} b c d + 24 \, a^{3} d^{2} + {\left (15 \, b^{3} c^{2} - 40 \, a b^{2} c d + 24 \, a^{2} b d^{2}\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 (2 \, b^{3} d^{2} x^{5} + 3 \, {\left (3 \, b^{3} c d - 2 \, a b^{2} d^{2}\right )} x^{3} - {\left (4 \, b^{3} c^{2} - 17 \, a b^{2} c d + 12 \, a^{2} b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, -\frac {{\left (15 \, a b^{2} c^{2} - 40 \, a^{2} b c d + 24 \, a^{3} d^{2} + {\left (15 \, b^{3} c^{2} - 40 \, a b^{2} c d + 24 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 2 \, {\left (a b^{2} c^{2} - 7 \, a^{2} b c d + 6 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 7 \, a b^{2} c d + 6 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) - {\left (2 \, b^{3} d^{2} x^{5} + 3 \, {\left (3 \, b^{3} c d - 2 \, a b^{2} d^{2}\right )} x^{3} - {\left (4 \, b^{3} c^{2} - 17 \, a b^{2} c d + 12 \, a^{2} b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{5} x^{2} + a 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 {x^{2} \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. 446 vs.
\(2 (163) = 326\).
time = 0.53, size = 446, normalized size = 2.29 \begin {gather*} \frac {1}{8} \, \sqrt {d x^{2} + c} {\left (\frac {2 \, d^{2} x^{2}}{b^{2}} + \frac {9 \, b^{7} c d^{3} - 8 \, a b^{6} d^{4}}{b^{9} d^{2}}\right )} x - \frac {{\left (15 \, b^{2} c^{2} \sqrt {d} - 40 \, a b c d^{\frac {3}{2}} + 24 \, a^{2} d^{\frac {5}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{16 \, b^{4}} - \frac {{\left (b^{3} c^{3} \sqrt {d} - 8 \, a b^{2} c^{2} d^{\frac {3}{2}} + 13 \, a^{2} b c d^{\frac {5}{2}} - 6 \, a^{3} 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^{4}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c^{3} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{2} d^{\frac {3}{2}} + 5 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac {5}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} d^{\frac {7}{2}} - b^{3} c^{4} \sqrt {d} + 2 \, a b^{2} c^{3} d^{\frac {3}{2}} - a^{2} 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^{4}} \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 {x^2\,{\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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