Optimal. Leaf size=543 \[ \frac {\left (a+b x+c x^2\right )^{7/2} \left (10240 a^2 B c^2-14 c x \left (2376 a A c^2-3380 a b B c-3146 A b^2 c+2145 b^3 B\right )+39688 a A b c^2-42900 a b^2 B c-28314 A b^3 c+19305 b^4 B\right )}{887040 c^5}+\frac {\left (b^2-4 a c\right )^3 \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{524288 c^{17/2}}-\frac {\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{262144 c^8}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{98304 c^7}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{30720 c^6}+\frac {x^2 \left (a+b x+c x^2\right )^{7/2} \left (-160 a B c-286 A b c+195 b^2 B\right )}{3960 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{7/2} (15 b B-22 A c)}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.72, antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {832, 779, 612, 621, 206} \[ \frac {\left (a+b x+c x^2\right )^{7/2} \left (10240 a^2 B c^2-14 c x \left (2376 a A c^2-3380 a b B c-3146 A b^2 c+2145 b^3 B\right )+39688 a A b c^2-42900 a b^2 B c-28314 A b^3 c+19305 b^4 B\right )}{887040 c^5}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{30720 c^6}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{98304 c^7}-\frac {\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right )}{262144 c^8}+\frac {\left (b^2-4 a c\right )^3 \left (-96 a^2 A c^3+240 a^2 b B c^2+528 a A b^2 c^2-520 a b^3 B c-286 A b^4 c+195 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{524288 c^{17/2}}+\frac {x^2 \left (a+b x+c x^2\right )^{7/2} \left (-160 a B c-286 A b c+195 b^2 B\right )}{3960 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{7/2} (15 b B-22 A c)}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int x^4 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\int x^3 \left (-4 a B-\frac {1}{2} (15 b B-22 A c) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{11 c}\\ &=-\frac {(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\int x^2 \left (\frac {3}{2} a (15 b B-22 A c)+\frac {1}{4} \left (195 b^2 B-286 A b c-160 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{110 c^2}\\ &=\frac {\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac {(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\int x \left (-\frac {1}{2} a \left (195 b^2 B-286 A b c-160 a B c\right )-\frac {1}{8} \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{990 c^3}\\ &=\frac {\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac {(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}-\frac {\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{2560 c^5}\\ &=-\frac {\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{30720 c^6}+\frac {\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac {(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}+\frac {\left (\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12288 c^6}\\ &=\frac {\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{98304 c^7}-\frac {\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{30720 c^6}+\frac {\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac {(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{65536 c^7}\\ &=-\frac {\left (b^2-4 a c\right )^2 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{262144 c^8}+\frac {\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{98304 c^7}-\frac {\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{30720 c^6}+\frac {\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac {(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}+\frac {\left (\left (b^2-4 a c\right )^3 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{524288 c^8}\\ &=-\frac {\left (b^2-4 a c\right )^2 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{262144 c^8}+\frac {\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{98304 c^7}-\frac {\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{30720 c^6}+\frac {\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac {(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}+\frac {\left (\left (b^2-4 a c\right )^3 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{262144 c^8}\\ &=-\frac {\left (b^2-4 a c\right )^2 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{262144 c^8}+\frac {\left (b^2-4 a c\right ) \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{98304 c^7}-\frac {\left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{30720 c^6}+\frac {\left (195 b^2 B-286 A b c-160 a B c\right ) x^2 \left (a+b x+c x^2\right )^{7/2}}{3960 c^3}-\frac {(15 b B-22 A c) x^3 \left (a+b x+c x^2\right )^{7/2}}{220 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{7/2}}{11 c}+\frac {\left (19305 b^4 B-28314 A b^3 c-42900 a b^2 B c+39688 a A b c^2+10240 a^2 B c^2-14 c \left (2145 b^3 B-3146 A b^2 c-3380 a b B c+2376 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{887040 c^5}+\frac {\left (b^2-4 a c\right )^3 \left (195 b^5 B-286 A b^4 c-520 a b^3 B c+528 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{524288 c^{17/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.01, size = 386, normalized size = 0.71 \[ \frac {\frac {11 \left (96 a^2 A c^3-240 a^2 b B c^2-528 a A b^2 c^2+520 a b^3 B c+286 A b^4 c-195 b^5 B\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (11 c x^2-20 a\right )+32 b c^2 x \left (13 a+8 c x^2\right )+15 b^4-40 b^3 c x\right )-15 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{7864320 c^{15/2}}+\frac {x^2 (a+x (b+c x))^{7/2} \left (-160 a B c-286 A b c+195 b^2 B\right )}{360 c^2}+\frac {(a+x (b+c x))^{7/2} \left (572 b^2 c (77 A c x-75 a B)+8 a b c^2 (4961 A+5915 B x)+16 a c^2 (640 a B-2079 A c x)-858 b^3 c (33 A+35 B x)+19305 b^4 B\right )}{80640 c^4}+\frac {x^3 (a+x (b+c x))^{7/2} (22 A c-15 b B)}{20 c}+B x^4 (a+x (b+c x))^{7/2}}{11 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.07, size = 1775, normalized size = 3.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 908, normalized size = 1.67 \[ \frac {1}{908328960} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, {\left (18 \, {\left (20 \, B c^{2} x + \frac {45 \, B b c^{11} + 22 \, A c^{12}}{c^{10}}\right )} x + \frac {465 \, B b^{2} c^{10} + 920 \, B a c^{11} + 902 \, A b c^{11}}{c^{10}}\right )} x + \frac {15 \, B b^{3} c^{9} + 17300 \, B a b c^{10} + 8426 \, A b^{2} c^{10} + 16632 \, A a c^{11}}{c^{10}}\right )} x - \frac {225 \, B b^{4} c^{8} - 1620 \, B a b^{2} c^{9} - 330 \, A b^{3} c^{9} - 144640 \, B a^{2} c^{10} - 279928 \, A a b c^{10}}{c^{10}}\right )} x + \frac {975 \, B b^{5} c^{7} - 8000 \, B a b^{3} c^{8} - 1430 \, A b^{4} c^{8} + 17040 \, B a^{2} b c^{9} + 10560 \, A a b^{2} c^{9} + 687456 \, A a^{2} c^{10}}{c^{10}}\right )} x - \frac {2145 \, B b^{6} c^{6} - 19760 \, B a b^{4} c^{7} - 3146 \, A b^{5} c^{7} + 53040 \, B a^{2} b^{2} c^{8} + 26400 \, A a b^{3} c^{8} - 30720 \, B a^{3} c^{9} - 58080 \, A a^{2} b c^{9}}{c^{10}}\right )} x + \frac {19305 \, B b^{7} c^{5} - 197340 \, B a b^{5} c^{6} - 28314 \, A b^{6} c^{6} + 637360 \, B a^{2} b^{3} c^{7} + 266200 \, A a b^{4} c^{7} - 617280 \, B a^{3} b c^{8} - 733920 \, A a^{2} b^{2} c^{8} + 443520 \, A a^{3} c^{9}}{c^{10}}\right )} x - \frac {45045 \, B b^{8} c^{4} - 506220 \, B a b^{6} c^{5} - 66066 \, A b^{7} c^{5} + 1908720 \, B a^{2} b^{4} c^{6} + 688248 \, A a b^{5} c^{6} - 2571840 \, B a^{3} b^{2} c^{7} - 2275680 \, A a^{2} b^{3} c^{7} + 655360 \, B a^{4} c^{8} + 2273920 \, A a^{3} b c^{8}}{c^{10}}\right )} x + \frac {225225 \, B b^{9} c^{3} - 2762760 \, B a b^{7} c^{4} - 330330 \, A b^{8} c^{4} + 11911680 \, B a^{2} b^{5} c^{5} + 3781008 \, A a b^{6} c^{5} - 20507520 \, B a^{3} b^{3} c^{6} - 14572800 \, A a^{2} b^{4} c^{6} + 10684160 \, B a^{4} b c^{7} + 20176640 \, A a^{3} b^{2} c^{7} - 5322240 \, A a^{4} c^{8}}{c^{10}}\right )} x - \frac {675675 \, B b^{10} c^{2} - 9009000 \, B a b^{8} c^{3} - 990990 \, A b^{9} c^{3} + 43834560 \, B a^{2} b^{6} c^{4} + 12400080 \, A a b^{7} c^{4} - 92062080 \, B a^{3} b^{4} c^{5} - 54730368 \, A a^{2} b^{5} c^{5} + 73201920 \, B a^{4} b^{2} c^{6} + 96940800 \, A a^{3} b^{3} c^{6} - 10485760 \, B a^{5} c^{7} - 52349440 \, A a^{4} b c^{7}}{c^{10}}\right )} - \frac {{\left (195 \, B b^{11} - 2860 \, B a b^{9} c - 286 \, A b^{10} c + 15840 \, B a^{2} b^{7} c^{2} + 3960 \, A a b^{8} c^{2} - 40320 \, B a^{3} b^{5} c^{3} - 20160 \, A a^{2} b^{6} c^{3} + 44800 \, B a^{4} b^{3} c^{4} + 44800 \, A a^{3} b^{4} c^{4} - 15360 \, B a^{5} b c^{5} - 38400 \, A a^{4} b^{2} c^{5} + 6144 \, A a^{5} c^{6}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{524288 \, c^{\frac {17}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 1848, normalized size = 3.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________