Optimal. Leaf size=1916 \[ \frac {-3 a^2 c^2 x^2 b^6+6 c^3 x b^6+6 a^2 c d x b^6-2 a^3 c^2 x^3 b^5-3 a c^3 x^2 b^5+3 a^3 c d x^2 b^5-6 a^3 d^2 x b^5-6 a c^2 d x b^5+2 a^2 c^3 x^3 b^4+9 a^3 c^3 x^2 b^4+3 a^2 c^2 d x^2 b^4-18 a^3 c^2 d x b^4+4 a^4 c^3 x^3 b^3+3 a^2 c^4 x^2 b^3-9 a^4 c^2 d x^2 b^3-6 a^3 c^4 x b^3+18 a^4 c d^2 x b^3-6 a^2 c^3 d x b^3-4 a^3 c^4 x^3 b^2-6 a^4 c^4 x^2 b^2-3 a^3 c^3 d x^2 b^2+6 a^3 c^2 d^2 x b^2+24 a^4 c^3 d x b^2-2 a^5 c^4 x^3 b+6 a^5 c^3 d x^2 b-18 a^5 c^2 d^2 x b+2 a^4 c^5 x^3}{6 a^2 c^2 \left (a c-b^2\right )^3}+\frac {\sqrt {\frac {b+a x}{d+c x}} \left (15 c^2 d b^7+15 c^3 x b^7-4 a c d^2 b^6-10 a c^3 x^2 b^6+3 a^2 c^2 d b^6+3 a^2 c^3 x b^6-14 a c^2 d x b^6-3 a^2 d^3 b^5+8 a^2 c^3 x^3 b^5+4 a^3 c d^2 b^5-2 a^3 c^3 x^2 b^5+10 a^2 c^2 d x^2 b^5+12 a c^3 d b^5+12 a c^4 x b^5-a^2 c d^2 x b^5+2 a^3 c^2 d x b^5-15 a^4 d^3 b^4-8 a^4 c^3 x^3 b^4-40 a^2 c^2 d^2 b^4+8 a^2 c^4 x^2 b^4+2 a^4 c^2 d x^2 b^4-24 a^3 c^3 d b^4-24 a^3 c^4 x b^4-5 a^4 c d^2 x b^4-32 a^2 c^3 d x b^4+12 a^3 c d^3 b^3-16 a^3 c^4 x^3 b^3-8 a^4 c^2 d^2 b^3+16 a^4 c^4 x^2 b^3-8 a^3 c^3 d x^2 b^3-3 a^2 c^4 d b^3-3 a^2 c^5 x b^3+20 a^3 c^2 d^2 x b^3+8 a^4 c^3 d x b^3+48 a^5 c d^3 b^2+16 a^5 c^4 x^3 b^2-4 a^3 c^3 d^2 b^2+2 a^3 c^5 x^2 b^2-16 a^5 c^3 d x^2 b^2-3 a^4 c^4 d b^2-3 a^4 c^5 x b^2+16 a^5 c^2 d^2 x b^2-2 a^3 c^4 d x b^2+15 a^4 c^2 d^3 b+8 a^4 c^5 x^3 b+52 a^5 c^3 d^2 b-14 a^5 c^5 x^2 b-2 a^4 c^4 d x^2 b+5 a^4 c^3 d^2 x b+38 a^5 c^4 d x b-57 a^6 c^2 d^3-8 a^6 c^5 x^3+14 a^6 c^4 d x^2-35 a^6 c^3 d^2 x\right )}{24 a^3 c^2 \left (a c-b^2\right )^3}+\frac {\left (5 c^3 b^{10}+a^2 c^3 b^9-3 a c^2 d b^9+15 a c^4 b^8-a^2 c d^2 b^8+a^3 c^2 d b^8-9 a^3 c^4 b^7-a^3 d^3 b^7+3 a^4 c d^2 b^7-33 a^2 c^3 d b^7-5 a^2 c^5 b^6-5 a^5 d^3 b^6+13 a^3 c^2 d^2 b^6-a^4 c^3 d b^6-9 a^4 c^5 b^5+5 a^4 c d^3 b^5-11 a^5 c^2 d^2 b^5-13 a^3 c^4 d b^5+a^3 c^6 b^4+21 a^6 c d^3 b^4+33 a^4 c^3 d^2 b^4+39 a^5 c^4 d b^4+a^5 c^6 b^3-15 a^5 c^2 d^3 b^3+5 a^6 c^3 d^2 b^3+a^4 c^5 d b^3-35 a^7 c^2 d^3 b^2+3 a^5 c^4 d^2 b^2+9 a^6 c^5 d b^2-5 a^6 c^3 d^3 b-45 a^7 c^4 d^2 b+35 a^8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2} \left (a c-b^2\right )^4}+\frac {\left (c^4 b^9+a^3 c d^2 b^8-a c^3 d b^8-a^4 d^3 b^7-a^3 c^5 b^6-4 a^4 c^2 d^2 b^6-2 a^2 c^4 d b^6+4 a^5 c d^3 b^5+2 a^3 c^3 d^2 b^5+a^4 c^4 d b^5+6 a^5 c^3 d^2 b^4-6 a^6 c^2 d^3 b^3+a^4 c^4 d^2 b^3+2 a^5 c^5 d b^3-a^5 c^3 d^3 b^2-6 a^6 c^4 d^2 b^2+4 a^7 c^3 d^3 b\right ) \log (d+c x)}{a^3 c^3 \left (a c-b^2\right )^4}+\frac {2 \left (c b^{10}-a d b^9-a^3 c^2 b^7-2 a^2 c d b^7+2 a^3 d^2 b^6+a^4 c d b^6+a^4 c d^2 b^4+2 a^5 c^2 d b^4-a^5 d^3 b^3-2 a^6 c d^2 b^3-a^7 c^2 d^2 b+a^8 c d^3\right ) \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (a c-b^2\right )^4} \]
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Rubi [A] time = 4.68, antiderivative size = 1063, normalized size of antiderivative = 0.55, number of steps used = 9, number of rules used = 5, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {1647, 1629, 635, 208, 260} \begin {gather*} -\frac {\left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right ) (b c-a d)^3}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {\left (5 c^2 b^9+a c \left (15 c^2+2 b d\right ) b^7+a^2 \left (-5 c^4+b^3 c^2-18 b d c^2+b^2 d^2\right ) b^5-a^3 c \left (-2 d b^4+9 c^2 b^3+5 d^2 b^2+18 c^2 d b-c^4\right ) b^3-a^4 \left (-5 d^2 b^4+10 c^2 d b^3+9 c^4 b^2-15 c^2 d^2 b-2 c^4 d\right ) b^2+5 a^6 c^2 d \left (2 c^2+7 b d\right ) b+a^5 c \left (b c^4+5 d^2 c^2+30 b^2 d c^2-21 b^3 d^2\right ) b-35 a^7 c^3 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right ) (b c-a d)}{8 a^{7/2} c^{5/2} \left (b^2-a c\right )^4}-\frac {b \left (c^3 b^7+a^3 d^2 b^6-a^2 c \left (a c^3+2 d c^2+4 a^2 d^2\right ) b^4+6 a^5 c^2 d^2 b^2+a^4 c^3 d (2 a c+d) b-4 a^6 c^3 d^2\right ) \log \left (\frac {1}{d+c x}\right ) (b c-a d)}{a^3 c^3 \left (b^2-a c\right )^4}+\frac {2 \left (b^3-a^3 c\right ) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) (b c-a d)}{a^3 b \left (b^2-a c\right )^4}+\frac {(d+c x) \left (8 a \left (-\left (\left (c^3+2 a^2 d c\right ) b^6\right )+3 a^3 d^2 b^5+6 a^3 c^2 d b^4+a^2 c \left (a c^3+2 d c^2-8 a^2 d^2\right ) b^3-6 a^4 c^3 d b^2+6 a^5 c^2 d^2 b-a^4 c^3 d^2\right )-c \left (5 c^2 b^7+2 a c \left (2 c^2+b d\right ) b^5+a^2 \left (-c^4+b^3 c^2-16 b d c^2+b^2 d^2\right ) b^3-2 a^3 c \left (-d b^3+4 c^2 b^2+2 d^2 b+c^2 d\right ) b^2+2 a^5 c d \left (11 c^2+16 b d\right ) b-a^4 \left (b c^4-11 d^2 c^2+8 b^2 d c^2+11 b^3 d^2\right ) b-29 a^6 c^2 d^2\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}+\frac {6 a \left (4 b c d a^3-\left (3 d b^3+2 c^2 b^2+2 c^2 d\right ) a^2+b^2 c \left (b^2+d\right ) a+b^3 c^2\right ) (b c-a d)^2+c \left (19 c d a^4-b \left (7 c^2+13 b d\right ) a^3+b c \left (b^2-13 d\right ) a^2+b^2 \left (c^2+7 b d\right ) a+5 b^4 c\right ) \sqrt {\frac {b+a x}{d+c x}} (b c-a d)^2}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 260
Rule 635
Rule 1629
Rule 1647
Rubi steps
\begin {align*} \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx &=(2 (b c-a d)) \operatorname {Subst}\left (\int \frac {x \left (b-d x^2\right )^2 \left (-1+c x^3\right )}{(a-b x) \left (a-c x^2\right )^4} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )\\ &=-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {\frac {a \left (a^2-b\right ) (b c-a d)^2}{c \left (b^2-a c\right )}-\frac {\left (5 b^4 c^2+2 a b^3 c d-5 a^4 b d^2+2 a^3 c d \left (5 b^2+3 d\right )-a^2 b \left (5 b^2 c^2+12 c^2 d+b d^2\right )\right ) x}{c \left (b^2-a c\right )}-\frac {6 a (b c-a d)^2 x^2}{c}+6 a d^2 x^3+6 a d (2 b c-a d) x^4-6 a c d^2 x^6}{(a-b x) \left (a-c x^2\right )^3} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c}\\ &=-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {\frac {3 a (b c-a d) \left (a^2 b^3 c-3 b^4 c+a^3 b c^2+a b^2 c^2+3 a^3 b^2 d-a b^3 d-5 a^4 c d+3 a^2 b c d\right )}{\left (b^2-a c\right )^2}-\frac {3 \left (5 b^6 c^2-19 a^5 b c d^2+a b^4 c \left (c^2+2 b d\right )+a^4 d \left (26 b^2 c^2+13 b^3 d+8 c^2 d\right )-a^3 b^2 c \left (7 b c^2+14 b^2 d+3 d^2\right )+a^2 b^3 \left (b^2 c^2-14 c^2 d+b d^2\right )\right ) x}{\left (b^2-a c\right )^2}-48 a^2 d (b c-a d) x^2+24 a^2 c d^2 x^4}{(a-b x) \left (a-c x^2\right )^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^2}\\ &=\frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {-\frac {3 a c \left (11 b^7 c^2-19 a^6 c^2 d^2-2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (5 c^2+8 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2+b^3 d-c^2 d-2 b d^2\right )-a^2 b^3 \left (b^3 c^2-c^4+16 b c^2 d+b^2 d^2\right )+a^4 b \left (b c^4+8 b^2 c^2 d-5 b^3 d^2+5 c^2 d^2\right )\right )}{\left (b^2-a c\right )^3}-\frac {3 b c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) x}{\left (b^2-a c\right )^3}-48 a^3 c d^2 x^2}{(a-b x) \left (a-c x^2\right )} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{24 a^3 c^3}\\ &=\frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {(b c-a d) \operatorname {Subst}\left (\int \left (-\frac {48 c^3 \left (-b^3+a^3 c\right ) \left (b^3-a^2 d\right )^2}{\left (b^2-a c\right )^4 (-a+b x)}+\frac {3 c \left (5 b^9 c^2-35 a^7 c^3 d^2+a b^7 c \left (15 c^2+2 b d\right )+5 a^6 b c^2 d \left (2 c^2+7 b d\right )+a^2 b^5 \left (b^3 c^2-5 c^4-18 b c^2 d+b^2 d^2\right )-a^3 b^3 c \left (9 b^3 c^2-c^4-2 b^4 d+18 b c^2 d+5 b^2 d^2\right )+a^5 b c \left (b c^4+30 b^2 c^2 d-21 b^3 d^2+5 c^2 d^2\right )-a^4 b^2 \left (9 b^2 c^4+10 b^3 c^2 d-2 c^4 d-5 b^4 d^2-15 b c^2 d^2\right )+16 b \left (b^7 c^3+a^3 b^6 d^2+6 a^5 b^2 c^2 d^2-4 a^6 c^3 d^2+a^4 b c^3 d (2 a c+d)-a^2 b^4 c \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right ) x\right )}{\left (b^2-a c\right )^4 \left (a-c x^2\right )}\right ) \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{24 a^3 c^3}\\ &=\frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {2 \left (b^3-a^3 c\right ) (b c-a d) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (b^2-a c\right )^4}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {5 b^9 c^2-35 a^7 c^3 d^2+a b^7 c \left (15 c^2+2 b d\right )+5 a^6 b c^2 d \left (2 c^2+7 b d\right )+a^2 b^5 \left (b^3 c^2-5 c^4-18 b c^2 d+b^2 d^2\right )-a^3 b^3 c \left (9 b^3 c^2-c^4-2 b^4 d+18 b c^2 d+5 b^2 d^2\right )+a^5 b c \left (b c^4+30 b^2 c^2 d-21 b^3 d^2+5 c^2 d^2\right )-a^4 b^2 \left (9 b^2 c^4+10 b^3 c^2 d-2 c^4 d-5 b^4 d^2-15 b c^2 d^2\right )+16 b \left (b^7 c^3+a^3 b^6 d^2+6 a^5 b^2 c^2 d^2-4 a^6 c^3 d^2+a^4 b c^3 d (2 a c+d)-a^2 b^4 c \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right ) x}{a-c x^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^2 \left (b^2-a c\right )^4}\\ &=\frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {2 \left (b^3-a^3 c\right ) (b c-a d) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (b^2-a c\right )^4}+\frac {\left (2 b (b c-a d) \left (b^7 c^3+a^3 b^6 d^2+6 a^5 b^2 c^2 d^2-4 a^6 c^3 d^2+a^4 b c^3 d (2 a c+d)-a^2 b^4 c \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {x}{a-c x^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 c^2 \left (b^2-a c\right )^4}+\frac {\left ((b c-a d) \left (5 b^9 c^2-35 a^7 c^3 d^2+a b^7 c \left (15 c^2+2 b d\right )+5 a^6 b c^2 d \left (2 c^2+7 b d\right )+a^2 b^5 \left (b^3 c^2-5 c^4-18 b c^2 d+b^2 d^2\right )-a^3 b^3 c \left (9 b^3 c^2-c^4-2 b^4 d+18 b c^2 d+5 b^2 d^2\right )+a^5 b c \left (b c^4+30 b^2 c^2 d-21 b^3 d^2+5 c^2 d^2\right )-a^4 b^2 \left (9 b^2 c^4+10 b^3 c^2 d-2 c^4 d-5 b^4 d^2-15 b c^2 d^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-c x^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^2 \left (b^2-a c\right )^4}\\ &=\frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {(b c-a d) \left (5 b^9 c^2-35 a^7 c^3 d^2+a b^7 c \left (15 c^2+2 b d\right )+5 a^6 b c^2 d \left (2 c^2+7 b d\right )+a^2 b^5 \left (b^3 c^2-5 c^4-18 b c^2 d+b^2 d^2\right )-a^3 b^3 c \left (9 b^3 c^2-c^4-2 b^4 d+18 b c^2 d+5 b^2 d^2\right )+a^5 b c \left (b c^4+30 b^2 c^2 d-21 b^3 d^2+5 c^2 d^2\right )-a^4 b^2 \left (9 b^2 c^4+10 b^3 c^2 d-2 c^4 d-5 b^4 d^2-15 b c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2} \left (b^2-a c\right )^4}-\frac {b (b c-a d) \left (b^7 c^3+a^3 b^6 d^2+6 a^5 b^2 c^2 d^2-4 a^6 c^3 d^2+a^4 b c^3 d (2 a c+d)-a^2 b^4 c \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right ) \log \left (\frac {1}{d+c x}\right )}{a^3 c^3 \left (b^2-a c\right )^4}+\frac {2 \left (b^3-a^3 c\right ) (b c-a d) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (b^2-a c\right )^4}\\ \end {align*}
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Mathematica [A] time = 5.33, size = 1379, normalized size = 0.72 \begin {gather*} \frac {1}{24} (a d-b c) \left (-\frac {8 \left (a^2-b\right ) \sqrt {\frac {b+a x}{d+c x}} (d+c x)^3}{a c^2 \left (a c-b^2\right ) (a d-b c)}+\frac {8 (a b-c) (d+c x)^3}{c^3 \left (a c-b^2\right ) (b c-a d)}+\frac {12 \left (4 b c d a^3-\left (3 d b^3+2 c^2 b^2+2 c^2 d\right ) a^2+b^2 c \left (b^2+d\right ) a+b^3 c^2\right ) (d+c x)^2}{a c^3 \left (b^2-a c\right )^2 (a d-b c)}+\frac {12 \left (4 c d a^3-b \left (2 c^2+3 b d\right ) a^2+b c \left (b^2-3 d\right ) a+b^2 \left (c^2+2 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}} (d+c x)^2}{a c^2 \left (b^2-a c\right )^2 (a d-b c)}+\frac {24 \left (\left (c^3+2 a^2 d c\right ) b^6-3 a^3 d^2 b^5-6 a^3 c^2 d b^4+a^2 c \left (-a c^3-2 d c^2+8 a^2 d^2\right ) b^3+6 a^4 c^3 d b^2-6 a^5 c^2 d^2 b+a^4 c^3 d^2\right ) (d+c x)}{a^2 c^3 \left (a c-b^2\right )^3 (a d-b c)}+\frac {24 \left (-6 c^2 d^2 a^5-3 b^4 d^2 a^3+3 b c^2 d (2 a c+d) a^3-3 b^3 c d (2 a c+d) a^2+b^2 c \left (-a c^3-2 d c^2+8 a^2 d^2\right ) a^2+b^5 \left (c^3+2 a^2 d c+a d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}} (d+c x)}{a^2 c^2 \left (a c-b^2\right )^3 (a d-b c)}+\frac {24 \left (6 c^2 d^2 a^5+3 b^4 d^2 a^3-3 b c^2 d (2 a c+d) a^3+3 b^3 c d (2 a c+d) a^2+b^2 c \left (a c^3+2 d c^2-8 a^2 d^2\right ) a^2-b^5 \left (c^3+2 a^2 d c+a d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{a^{5/2} c^{5/2} \left (b^2-a c\right )^3}-\frac {5 \left (a^2-b\right ) \left (3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right ) (b c-a d)^2+\sqrt {a} \sqrt {c} \sqrt {\frac {b+a x}{d+c x}} (d+c x) (-3 b c+2 a x c+5 a d)\right )}{a^{7/2} c^{5/2} \left (a c-b^2\right )}+\frac {18 \left (4 c d a^3-b \left (2 c^2+3 b d\right ) a^2+b c \left (b^2-3 d\right ) a+b^2 \left (c^2+2 b d\right )\right ) \left (\sqrt {a} \sqrt {c} \sqrt {\frac {b+a x}{d+c x}} (d+c x)+(a d-b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )\right )}{a^{5/2} c^{5/2} \left (b^2-a c\right )^2}-\frac {48 \left (b^3-a^3 c\right ) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (b^2-a c\right )^4}+\frac {24 \left (c^3 b^7+a^3 d^2 b^6-a^2 c \left (a c^3+2 d c^2+4 a^2 d^2\right ) b^4+6 a^5 c^2 d^2 b^2+a^4 c^3 d (2 a c+d) b-4 a^6 c^3 d^2\right ) \log \left (\sqrt {a}-\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 \left (b-\sqrt {a} \sqrt {c}\right )^4 \left (b+\sqrt {a} \sqrt {c}\right )^3 c^3}-\frac {24 \left (c^3 b^7+a^3 d^2 b^6-a^2 c \left (a c^3+2 d c^2+4 a^2 d^2\right ) b^4+6 a^5 c^2 d^2 b^2+a^4 c^3 d (2 a c+d) b-4 a^6 c^3 d^2\right ) \log \left (\sqrt {a}+\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}\right )}{\left (b+\sqrt {a} \sqrt {c}\right )^4 \left (a^{3/2} c^{3/2}-a b c\right )^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 3.46, size = 4104, normalized size = 2.14 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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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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maple [B] time = 0.55, size = 384279, normalized size = 200.56
method | result | size |
default | \(\text {Expression too large to display}\) | \(384279\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 2606, normalized size = 1.36
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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