Optimal. Leaf size=443 \[ \frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right ) \tanh ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right ) \tanh ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \]
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Rubi [A]
time = 0.97, antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {913, 990, 1048,
739, 212} \begin {gather*} \frac {c \left (-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac {d^2 x \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {c \left (-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac {d^2 x \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac {d^2 \left (b-x \left (a d^2+c\right )\right )}{\sqrt {1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 913
Rule 990
Rule 1048
Rubi steps
\begin {align*} \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )} \, dx &=\int \frac {1}{\left (a+b x+c x^2\right ) \left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}-\frac {\int \frac {2 d^2 \left (c^2-b^2 d^2+a c d^2\right )-2 b c d^4 x}{\left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}} \, dx}{2 d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}+\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}-\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 c^2-\left (b+\sqrt {b^2-4 a c}\right )^2 d^2-x^2} \, dx,x,\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {1-d^2 x^2}}\right )}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}+\frac {\left (c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 c^2-\left (b-\sqrt {b^2-4 a c}\right )^2 d^2-x^2} \, dx,x,\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {1-d^2 x^2}}\right )}{\sqrt {b^2-4 a c} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=\frac {d^2 \left (b-\left (c+a d^2\right ) x\right )}{\left (b^2 d^2-\left (c+a d^2\right )^2\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right ) \tanh ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}-\frac {c \left (2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right ) \tanh ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.68, size = 1546, normalized size = 3.49 \begin {gather*} \frac {4 d^2 \left (-b+\left (c+a d^2\right ) x\right ) \sqrt {1-d^2 x^2}+2 c^2 \sqrt {-d^2} \left (c+a d^2\right ) (-1+d x) (1+d x) \text {RootSum}\left [c^2-4 c^2 \text {$\#$1}+4 b^2 d^2 \text {$\#$1}-8 a c d^2 \text {$\#$1}+6 c^2 \text {$\#$1}^2-8 b^2 d^2 \text {$\#$1}^2+16 a c d^2 \text {$\#$1}^2+16 a^2 d^4 \text {$\#$1}^2-4 c^2 \text {$\#$1}^3+4 b^2 d^2 \text {$\#$1}^3-8 a c d^2 \text {$\#$1}^3+c^2 \text {$\#$1}^4\&,\frac {\log \left (-1+2 d^2 x^2+2 \sqrt {-d^2} x \sqrt {1-d^2 x^2}+\text {$\#$1}\right )}{-c^2+b^2 d^2-2 a c d^2+3 c^2 \text {$\#$1}-4 b^2 d^2 \text {$\#$1}+8 a c d^2 \text {$\#$1}+8 a^2 d^4 \text {$\#$1}-3 c^2 \text {$\#$1}^2+3 b^2 d^2 \text {$\#$1}^2-6 a c d^2 \text {$\#$1}^2+c^2 \text {$\#$1}^3}\&\right ]-4 \sqrt {-d^2} \left (c^3+3 a c^2 d^2-2 a b^2 d^4+2 a^2 c d^4\right ) (-1+d x) (1+d x) \text {RootSum}\left [c^2-4 c^2 \text {$\#$1}+4 b^2 d^2 \text {$\#$1}-8 a c d^2 \text {$\#$1}+6 c^2 \text {$\#$1}^2-8 b^2 d^2 \text {$\#$1}^2+16 a c d^2 \text {$\#$1}^2+16 a^2 d^4 \text {$\#$1}^2-4 c^2 \text {$\#$1}^3+4 b^2 d^2 \text {$\#$1}^3-8 a c d^2 \text {$\#$1}^3+c^2 \text {$\#$1}^4\&,\frac {\log \left (-1+2 d^2 x^2+2 \sqrt {-d^2} x \sqrt {1-d^2 x^2}+\text {$\#$1}\right ) \text {$\#$1}}{-c^2+b^2 d^2-2 a c d^2+3 c^2 \text {$\#$1}-4 b^2 d^2 \text {$\#$1}+8 a c d^2 \text {$\#$1}+8 a^2 d^4 \text {$\#$1}-3 c^2 \text {$\#$1}^2+3 b^2 d^2 \text {$\#$1}^2-6 a c d^2 \text {$\#$1}^2+c^2 \text {$\#$1}^3}\&\right ]+2 c^2 \sqrt {-d^2} \left (c+a d^2\right ) (-1+d x) (1+d x) \text {RootSum}\left [c^2-4 c^2 \text {$\#$1}+4 b^2 d^2 \text {$\#$1}-8 a c d^2 \text {$\#$1}+6 c^2 \text {$\#$1}^2-8 b^2 d^2 \text {$\#$1}^2+16 a c d^2 \text {$\#$1}^2+16 a^2 d^4 \text {$\#$1}^2-4 c^2 \text {$\#$1}^3+4 b^2 d^2 \text {$\#$1}^3-8 a c d^2 \text {$\#$1}^3+c^2 \text {$\#$1}^4\&,\frac {\log \left (-1+2 d^2 x^2+2 \sqrt {-d^2} x \sqrt {1-d^2 x^2}+\text {$\#$1}\right ) \text {$\#$1}^2}{-c^2+b^2 d^2-2 a c d^2+3 c^2 \text {$\#$1}-4 b^2 d^2 \text {$\#$1}+8 a c d^2 \text {$\#$1}+8 a^2 d^4 \text {$\#$1}-3 c^2 \text {$\#$1}^2+3 b^2 d^2 \text {$\#$1}^2-6 a c d^2 \text {$\#$1}^2+c^2 \text {$\#$1}^3}\&\right ]+b d^2 (-1+d x) (1+d x) \text {RootSum}\left [c^2-4 c^2 \text {$\#$1}^2+4 b^2 d^2 \text {$\#$1}^2-8 a c d^2 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-8 b^2 d^2 \text {$\#$1}^4+16 a c d^2 \text {$\#$1}^4+16 a^2 d^4 \text {$\#$1}^4-4 c^2 \text {$\#$1}^6+4 b^2 d^2 \text {$\#$1}^6-8 a c d^2 \text {$\#$1}^6+c^2 \text {$\#$1}^8\&,\frac {-c^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right )+7 c^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-4 b^2 d^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+8 a c d^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-7 c^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+4 b^2 d^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4-8 a c d^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+c^2 \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^6}{-c^2 \text {$\#$1}+b^2 d^2 \text {$\#$1}-2 a c d^2 \text {$\#$1}+3 c^2 \text {$\#$1}^3-4 b^2 d^2 \text {$\#$1}^3+8 a c d^2 \text {$\#$1}^3+8 a^2 d^4 \text {$\#$1}^3-3 c^2 \text {$\#$1}^5+3 b^2 d^2 \text {$\#$1}^5-6 a c d^2 \text {$\#$1}^5+c^2 \text {$\#$1}^7}\&\right ]}{4 (c+d (-b+a d)) (c+d (b+a d)) (1-d x) (1+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.15, size = 11142, normalized size = 25.15
method | result | size |
default | \(\text {Expression too large to display}\) | \(11142\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 21628 vs.
\(2 (404) = 808\).
time = 28.51, size = 21628, normalized size = 48.82 \begin {gather*} \text {Too large to display} \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 {1}{\left (- d x + 1\right )^{\frac {3}{2}} \left (d x + 1\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {1}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}\,\left (c\,x^2+b\,x+a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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