Optimal. Leaf size=939 \[ -\frac {\left (b \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-\left (2 c^4-d^2 \left (b^2+6 a^2 d^2\right ) c^2-\left (4 a^3 d^6+6 a b^2 d^4\right ) c+b^2 d^4 \left (2 b^2+a^2 d^2\right )\right ) x\right ) d^2}{\left (b^2-4 a c\right ) \left (a d^2-b d+c\right )^2 \left (a d^2+b d+c\right )^2 \sqrt {1-d^2 x^2}}+\frac {c \left (3 a b^3 \left (b+\sqrt {b^2-4 a c}\right ) d^6-2 a c^2 \left (7 b^2+5 \sqrt {b^2-4 a c} b-8 a^2 d^2\right ) d^4+b c \left (2 b^3+2 \sqrt {b^2-4 a c} b^2-17 a^2 d^2 b-11 a^2 \sqrt {b^2-4 a c} d^2\right ) d^4+24 a c^4 d^2-c^3 \left (9 b^2-\sqrt {b^2-4 a c} b-36 a^2 d^2\right ) d^2+4 c^5\right ) \tanh ^{-1}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (b \left (b+\sqrt {b^2-4 a c}\right ) d^4 \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-2 \left (3 a b^4 d^8+2 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+12 a c^4 d^4+2 c^5 d^2-c^3 \left (4 b^2 d^4-18 a^2 d^6\right )-4 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \tanh ^{-1}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2 d^2} \]
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Rubi [A] time = 11.84, antiderivative size = 938, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {899, 975, 1062, 1034, 725, 206} \[ -\frac {\left (b \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-\left (2 c^4-d^2 \left (b^2+6 a^2 d^2\right ) c^2-\left (4 a^3 d^6+6 a b^2 d^4\right ) c+b^2 d^4 \left (2 b^2+a^2 d^2\right )\right ) x\right ) d^2}{\left (b^2-4 a c\right ) \left (a d^2-b d+c\right )^2 \left (a d^2+b d+c\right )^2 \sqrt {1-d^2 x^2}}+\frac {c \left (3 a b^3 \left (b+\sqrt {b^2-4 a c}\right ) d^6-2 a c^2 \left (7 b^2+5 \sqrt {b^2-4 a c} b-8 a^2 d^2\right ) d^4+b c \left (2 b^3+2 \sqrt {b^2-4 a c} b^2-17 a^2 d^2 b-11 a^2 \sqrt {b^2-4 a c} d^2\right ) d^4+24 a c^4 d^2-c^3 \left (9 b^2-\sqrt {b^2-4 a c} b-36 a^2 d^2\right ) d^2+4 c^5\right ) \tanh ^{-1}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}-\frac {c \left (6 a b^4 d^8+4 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+24 a c^4 d^4-b \left (b+\sqrt {b^2-4 a c}\right ) \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) d^4+4 c^5 d^2-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \tanh ^{-1}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2 d^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 899
Rule 975
Rule 1034
Rule 1062
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 )^2} \, dx &=\int \frac {1}{\left (a+b x+c x^2\right )^2 \left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}}-\frac {\int \frac {-2 c^3-6 a c^2 d^2+a b^2 d^4+2 c d^2 \left (b^2-2 a^2 d^2\right )+b d^2 \left (c^2-2 b^2 d^2+7 a c d^2\right ) x+2 c d^2 \left (2 c^2-b^2 d^2+2 a c d^2\right ) x^2}{\left (a+b x+c x^2\right ) \left (1-d^2 x^2\right )^{3/2}} \, dx}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=-\frac {d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2 \sqrt {1-d^2 x^2}}-\frac {b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}}-\frac {\int \frac {2 \left (2 c^5 d^2+12 a c^4 d^4+3 a b^4 d^8+2 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-\frac {1}{2} c^3 \left (8 b^2 d^4-36 a^2 d^6\right )-4 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )+2 b c d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right ) x}{\left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}} \, dx}{2 \left (b^2-4 a c\right ) d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}\\ &=-\frac {d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2 \sqrt {1-d^2 x^2}}-\frac {b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}}-\frac {\left (c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b-\sqrt {b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right )^{3/2} d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}+\frac {\left (c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b+\sqrt {b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right )^{3/2} d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}\\ &=-\frac {d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2 \sqrt {1-d^2 x^2}}-\frac {b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}}+\frac {\left (c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b-\sqrt {b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \operatorname {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 )}{\left (b^2-4 a c\right )^{3/2} d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}-\frac {\left (c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b+\sqrt {b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \operatorname {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 )}{\left (b^2-4 a c\right )^{3/2} d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}\\ &=-\frac {d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2 \sqrt {1-d^2 x^2}}-\frac {b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b-\sqrt {b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\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} \left (b^2-4 a c\right )^{3/2} d^2 \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 )^2}-\frac {c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b+\sqrt {b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\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} \left (b^2-4 a c\right )^{3/2} d^2 \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 )^2}\\ \end {align*}
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Mathematica [A] time = 9.22, size = 890, normalized size = 0.95 \[ \frac {\frac {\sqrt {2} \left (-3 a b \left (b+\sqrt {b^2-4 a c}\right ) d^4+20 a c^2 d^2-c \left (7 b^2-3 \sqrt {b^2-4 a c} b-16 a^2 d^2\right ) d^2+4 c^3\right ) \tanh ^{-1}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right ) c^3}{\sqrt {b^2-4 a c} \left (2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2\right )^{3/2}}-\frac {\sqrt {2} \left (-3 a b \left (b-\sqrt {b^2-4 a c}\right ) d^4+20 a c^2 d^2-c \left (7 b^2+3 \sqrt {b^2-4 a c} b-16 a^2 d^2\right ) d^2+4 c^3\right ) \tanh ^{-1}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right ) c^3}{\sqrt {b^2-4 a c} \left (2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2\right )^{3/2}}-\frac {\left (3 b \left (c-a d^2\right ) d^2+\frac {-3 a b^2 d^4+16 a^2 c d^4+20 a c^2 d^2-7 b^2 c d^2+4 c^3}{\sqrt {b^2-4 a c}}\right ) \left (2 c-\left (b-\sqrt {b^2-4 a c}\right ) d^2 x\right ) c}{\left (4 c^2-\left (b-\sqrt {b^2-4 a c}\right )^2 d^2\right ) \sqrt {1-d^2 x^2}}-\frac {\left (3 b d^2 \left (c-a d^2\right )-\frac {-3 a b^2 d^4+16 a^2 c d^4+20 a c^2 d^2-7 b^2 c d^2+4 c^3}{\sqrt {b^2-4 a c}}\right ) \left (2 c-\left (b+\sqrt {b^2-4 a c}\right ) d^2 x\right ) c}{\left (4 c^2-\left (b+\sqrt {b^2-4 a c}\right )^2 d^2\right ) \sqrt {1-d^2 x^2}}+\frac {2 d^2 \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\sqrt {1-d^2 x^2}}}{\left (b^2-4 a c\right ) \left (\left (a d^2+c\right )^2-b^2 d^2\right )}-\frac {-d^2 b^3+c \left (3 a d^2+c\right ) b+c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (\left (a d^2+c\right )^2-b^2 d^2\right ) \left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.14, size = 108974, normalized size = 116.05 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x + a\right )}^{2} {\left (d x + 1\right )}^{\frac {3}{2}} {\left (-d x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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