Optimal. Leaf size=484 \[ \frac {b \left (25 c^2 \left (e^2 f+2 d e g+d^2 h\right )+12 e (e h+2 d i)\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b \left (5 e^2 i+9 c^2 \left (e^2 g+2 d e h+d^2 i\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e (e h+2 d i) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^2 i x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (225 c^4 d^2 f+50 c^2 \left (e^2 f+2 d e g+d^2 h\right )+24 e (e h+2 d i)\right )+75 \left (24 c^4 d (2 e f+d g)+5 e^2 i+9 c^2 \left (e^2 g+2 d e h+d^2 i\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}-\frac {b \left (24 c^4 d (2 e f+d g)+5 e^2 i+9 c^2 \left (e^2 g+2 d e h+d^2 i\right )\right ) \text {ArcSin}(c x)}{96 c^6}+d^2 f x (a+b \text {ArcSin}(c x))+\frac {1}{2} d (2 e f+d g) x^2 (a+b \text {ArcSin}(c x))+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 (a+b \text {ArcSin}(c x))+\frac {1}{4} \left (e^2 g+2 d e h+d^2 i\right ) x^4 (a+b \text {ArcSin}(c x))+\frac {1}{5} e (e h+2 d i) x^5 (a+b \text {ArcSin}(c x))+\frac {1}{6} e^2 i x^6 (a+b \text {ArcSin}(c x)) \]
[Out]
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Rubi [A]
time = 1.40, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4833, 12, 1823,
794, 222} \begin {gather*} \frac {1}{3} x^3 (a+b \text {ArcSin}(c x)) \left (d^2 h+2 d e g+e^2 f\right )+\frac {1}{4} x^4 (a+b \text {ArcSin}(c x)) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x (a+b \text {ArcSin}(c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \text {ArcSin}(c x))+\frac {1}{5} e x^5 (2 d i+e h) (a+b \text {ArcSin}(c x))+\frac {1}{6} e^2 i x^6 (a+b \text {ArcSin}(c x))-\frac {b \text {ArcSin}(c x) \left (24 c^4 d (d g+2 e f)+9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+5 e^2 i\right )}{96 c^6}+\frac {b x^3 \sqrt {1-c^2 x^2} \left (e^2 \left (\frac {5 i}{c^2}+9 g\right )+9 d^2 i+18 d e h\right )}{144 c}+\frac {b e x^4 \sqrt {1-c^2 x^2} (2 d i+e h)}{25 c}+\frac {b e^2 i x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e (2 d i+e h)\right )}{225 c^3}+\frac {b \sqrt {1-c^2 x^2} \left (75 x \left (24 c^4 d (d g+2 e f)+9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+5 e^2 i\right )+32 \left (225 c^4 d^2 f+50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+24 e (2 d i+e h)\right )\right )}{7200 c^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 222
Rule 794
Rule 1823
Rule 4833
Rubi steps
\begin {align*} \int (d+e x)^2 \left (f+g x+h x^2+107 x^3\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (5 d^2 (12 f+x (6 g+x (4 h+321 x)))+2 d e x (30 f+x (20 g+3 x (5 h+428 x)))+e^2 x^2 (20 f+x (15 g+2 x (6 h+535 x)))\right )}{60 \sqrt {1-c^2 x^2}} \, dx\\ &=d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{60} (b c) \int \frac {x \left (5 d^2 (12 f+x (6 g+x (4 h+321 x)))+2 d e x (30 f+x (20 g+3 x (5 h+428 x)))+e^2 x^2 (20 f+x (15 g+2 x (6 h+535 x)))\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {107 b e^2 x^5 \sqrt {1-c^2 x^2}}{36 c}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-360 c^2 d^2 f-180 c^2 d (2 e f+d g) x-120 c^2 \left (e^2 f+2 d e g+d^2 h\right ) x^2-10 \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3-72 c^2 e (214 d+e h) x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{360 c}\\ &=\frac {b e (214 d+e h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {107 b e^2 x^5 \sqrt {1-c^2 x^2}}{36 c}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (1800 c^4 d^2 f+900 c^4 d (2 e f+d g) x+24 c^2 \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2+50 c^2 \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3\right )}{\sqrt {1-c^2 x^2}} \, dx}{1800 c^3}\\ &=\frac {b \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e (214 d+e h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {107 b e^2 x^5 \sqrt {1-c^2 x^2}}{36 c}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-7200 c^6 d^2 f-150 c^2 \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x-96 c^4 \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{7200 c^5}\\ &=\frac {b \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e (214 d+e h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {107 b e^2 x^5 \sqrt {1-c^2 x^2}}{36 c}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (96 c^4 \left (4 d e \left (1284+25 c^2 g\right )+2 e^2 \left (25 c^2 f+12 h\right )+25 d^2 \left (9 c^4 f+2 c^2 h\right )\right )+450 c^4 \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{21600 c^7}\\ &=\frac {b \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e (214 d+e h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {107 b e^2 x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (4 d e \left (1284+25 c^2 g\right )+2 e^2 \left (25 c^2 f+12 h\right )+25 d^2 \left (9 c^4 f+2 c^2 h\right )\right )+75 \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac {b \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e (214 d+e h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {107 b e^2 x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (4 d e \left (1284+25 c^2 g\right )+2 e^2 \left (25 c^2 f+12 h\right )+25 d^2 \left (9 c^4 f+2 c^2 h\right )\right )+75 \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}-\frac {b \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) \sin ^{-1}(c x)}{96 c^6}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 431, normalized size = 0.89 \begin {gather*} a d^2 f x+\frac {1}{2} a d (2 e f+d g) x^2+\frac {1}{3} a \left (e^2 f+2 d e g+d^2 h\right ) x^3+\frac {1}{4} a \left (e^2 g+2 d e h+d^2 i\right ) x^4+\frac {1}{5} a e (e h+2 d i) x^5+\frac {1}{6} a e^2 i x^6+\frac {b \sqrt {1-c^2 x^2} \left (3 e (256 e h+512 d i+125 e i x)+c^2 \left (25 d^2 (64 h+27 i x)+2 d e \left (1600 g+675 h x+384 i x^2\right )+e^2 \left (1600 f+x \left (675 g+384 h x+250 i x^2\right )\right )\right )+2 c^4 \left (25 d^2 (144 f+x (36 g+x (16 h+9 i x)))+2 d e x (900 f+x (400 g+9 x (25 h+16 i x)))+e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))\right )\right )}{7200 c^5}-\frac {b \left (24 c^4 d (2 e f+d g)+5 e^2 i+9 c^2 \left (e^2 g+2 d e h+d^2 i\right )\right ) \text {ArcSin}(c x)}{96 c^6}+\frac {1}{60} b x \left (5 d^2 (12 f+x (6 g+x (4 h+3 i x)))+2 d e x (30 f+x (20 g+3 x (5 h+4 i x)))+e^2 x^2 (20 f+x (15 g+2 x (6 h+5 i x)))\right ) \text {ArcSin}(c x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.11, size = 674, normalized size = 1.39
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {e^{2} i \,c^{6} x^{6}}{6}+\frac {\left (2 c d e i +c \,e^{2} h \right ) c^{5} x^{5}}{5}+\frac {\left (c^{2} d^{2} i +2 c^{2} d e h +c^{2} e^{2} g \right ) c^{4} x^{4}}{4}+\frac {\left (c^{3} d^{2} h +2 c^{3} d e g +c^{3} e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{4} d^{2} g +2 c^{4} d e f \right ) c^{2} x^{2}}{2}+d^{2} c^{6} f x \right )}{c^{5}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} i \,c^{6} x^{6}}{6}+\frac {2 \arcsin \left (c x \right ) c^{6} d e i \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{6} e^{2} h \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{6} d^{2} i \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{6} d e h \,x^{4}}{2}+\frac {\arcsin \left (c x \right ) c^{6} e^{2} g \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{6} d^{2} h \,x^{3}}{3}+\frac {2 \arcsin \left (c x \right ) c^{6} d e g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{6} e^{2} f \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{6} d^{2} g \,x^{2}}{2}+\arcsin \left (c x \right ) c^{6} d e f \,x^{2}+\arcsin \left (c x \right ) d^{2} c^{6} f x -\frac {e^{2} i \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}-\frac {\left (24 c d e i +12 c \,e^{2} h \right ) \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{60}-\frac {\left (15 c^{2} d^{2} i +30 c^{2} d e h +15 c^{2} e^{2} g \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{60}-\frac {\left (20 c^{3} d^{2} h +40 c^{3} d e g +20 c^{3} e^{2} f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{60}-\frac {\left (30 c^{4} d^{2} g +60 c^{4} d e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{60}+d^{2} c^{5} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{5}}}{c}\) | \(674\) |
default | \(\frac {\frac {a \left (\frac {e^{2} i \,c^{6} x^{6}}{6}+\frac {\left (2 c d e i +c \,e^{2} h \right ) c^{5} x^{5}}{5}+\frac {\left (c^{2} d^{2} i +2 c^{2} d e h +c^{2} e^{2} g \right ) c^{4} x^{4}}{4}+\frac {\left (c^{3} d^{2} h +2 c^{3} d e g +c^{3} e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{4} d^{2} g +2 c^{4} d e f \right ) c^{2} x^{2}}{2}+d^{2} c^{6} f x \right )}{c^{5}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} i \,c^{6} x^{6}}{6}+\frac {2 \arcsin \left (c x \right ) c^{6} d e i \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{6} e^{2} h \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{6} d^{2} i \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{6} d e h \,x^{4}}{2}+\frac {\arcsin \left (c x \right ) c^{6} e^{2} g \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{6} d^{2} h \,x^{3}}{3}+\frac {2 \arcsin \left (c x \right ) c^{6} d e g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{6} e^{2} f \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{6} d^{2} g \,x^{2}}{2}+\arcsin \left (c x \right ) c^{6} d e f \,x^{2}+\arcsin \left (c x \right ) d^{2} c^{6} f x -\frac {e^{2} i \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}-\frac {\left (24 c d e i +12 c \,e^{2} h \right ) \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{60}-\frac {\left (15 c^{2} d^{2} i +30 c^{2} d e h +15 c^{2} e^{2} g \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{60}-\frac {\left (20 c^{3} d^{2} h +40 c^{3} d e g +20 c^{3} e^{2} f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{60}-\frac {\left (30 c^{4} d^{2} g +60 c^{4} d e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{60}+d^{2} c^{5} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{5}}}{c}\) | \(674\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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Maxima [A]
time = 0.48, size = 834, normalized size = 1.72 \begin {gather*} \frac {1}{5} \, a h x^{5} e^{2} + \frac {1}{6} i \, a x^{6} e^{2} + \frac {1}{2} \, a d h x^{4} e + \frac {2}{5} i \, a d x^{5} e + \frac {1}{3} \, a d^{2} h x^{3} + \frac {1}{4} i \, a d^{2} x^{4} + \frac {1}{4} \, a g x^{4} e^{2} + \frac {2}{3} \, a d g x^{3} e + \frac {1}{2} \, a d^{2} g x^{2} + \frac {1}{3} \, a f x^{3} e^{2} + a d f x^{2} e + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} h + a d^{2} f x + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d f e + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d g e + \frac {1}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d h e + \frac {1}{32} i \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2} f}{c} + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b f e^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b g e^{2} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b h e^{2} + \frac {2}{75} i \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e + \frac {1}{288} i \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.34, size = 612, normalized size = 1.26 \begin {gather*} \frac {4800 \, a c^{6} d^{2} h x^{3} + 3600 i \, a c^{6} d^{2} x^{4} + 7200 \, a c^{6} d^{2} g x^{2} + 14400 \, a c^{6} d^{2} f x + 240 \, {\left (12 \, a c^{6} h x^{5} + 10 i \, a c^{6} x^{6} + 15 \, a c^{6} g x^{4} + 20 \, a c^{6} f x^{3}\right )} e^{2} + 480 \, {\left (15 \, a c^{6} d h x^{4} + 12 i \, a c^{6} d x^{5} + 20 \, a c^{6} d g x^{3} + 30 \, a c^{6} d f x^{2}\right )} e - 15 \, {\left (160 i \, b c^{6} d^{2} h x^{3} - 120 \, b c^{6} d^{2} x^{4} + 240 i \, b c^{6} d^{2} g x^{2} + 480 i \, b c^{6} d^{2} f x - 120 i \, b c^{4} d^{2} g + 45 \, b c^{2} d^{2} + {\left (96 i \, b c^{6} h x^{5} - 80 \, b c^{6} x^{6} + 120 i \, b c^{6} g x^{4} + 160 i \, b c^{6} f x^{3} - 45 i \, b c^{2} g + 25 \, b\right )} e^{2} + 2 \, {\left (120 i \, b c^{6} d h x^{4} - 96 \, b c^{6} d x^{5} + 160 i \, b c^{6} d g x^{3} + 240 i \, b c^{6} d f x^{2} - 120 i \, b c^{4} d f - 45 i \, b c^{2} d h\right )} e\right )} \log \left (-2 \, c^{2} x^{2} - 2 \, \sqrt {c^{2} x^{2} - 1} c x + 1\right ) - 2 \, {\left (-800 i \, b c^{5} d^{2} h x^{2} + 450 \, b c^{5} d^{2} x^{3} - 7200 i \, b c^{5} d^{2} f - 1600 i \, b c^{3} d^{2} h + 225 \, {\left (-8 i \, b c^{5} d^{2} g + 3 \, b c^{3} d^{2}\right )} x + {\left (-288 i \, b c^{5} h x^{4} + 200 \, b c^{5} x^{5} - 1600 i \, b c^{3} f + 50 \, {\left (-9 i \, b c^{5} g + 5 \, b c^{3}\right )} x^{3} - 768 i \, b c h + 32 \, {\left (-25 i \, b c^{5} f - 12 i \, b c^{3} h\right )} x^{2} + 75 \, {\left (-9 i \, b c^{3} g + 5 \, b c\right )} x\right )} e^{2} + 2 \, {\left (-450 i \, b c^{5} d h x^{3} + 288 \, b c^{5} d x^{4} - 1600 i \, b c^{3} d g + 768 \, b c d + 32 \, {\left (-25 i \, b c^{5} d g + 12 \, b c^{3} d\right )} x^{2} + 225 \, {\left (-8 i \, b c^{5} d f - 3 i \, b c^{3} d h\right )} x\right )} e\right )} \sqrt {c^{2} x^{2} - 1}}{14400 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1197 vs.
\(2 (474) = 948\).
time = 0.80, size = 1197, normalized size = 2.47 \begin {gather*} \begin {cases} a d^{2} f x + \frac {a d^{2} g x^{2}}{2} + \frac {a d^{2} h x^{3}}{3} + \frac {a d^{2} i x^{4}}{4} + a d e f x^{2} + \frac {2 a d e g x^{3}}{3} + \frac {a d e h x^{4}}{2} + \frac {2 a d e i x^{5}}{5} + \frac {a e^{2} f x^{3}}{3} + \frac {a e^{2} g x^{4}}{4} + \frac {a e^{2} h x^{5}}{5} + \frac {a e^{2} i x^{6}}{6} + b d^{2} f x \operatorname {asin}{\left (c x \right )} + \frac {b d^{2} g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b d^{2} h x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d^{2} i x^{4} \operatorname {asin}{\left (c x \right )}}{4} + b d e f x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 b d e g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d e h x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {2 b d e i x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b e^{2} f x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e^{2} g x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b e^{2} h x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b e^{2} i x^{6} \operatorname {asin}{\left (c x \right )}}{6} + \frac {b d^{2} f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d^{2} g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d^{2} h x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b d^{2} i x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b d e f x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {2 b d e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b d e h x^{3} \sqrt {- c^{2} x^{2} + 1}}{8 c} + \frac {2 b d e i x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b e^{2} f x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b e^{2} g x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b e^{2} h x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b e^{2} i x^{5} \sqrt {- c^{2} x^{2} + 1}}{36 c} - \frac {b d^{2} g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {b d e f \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {2 b d^{2} h \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b d^{2} i x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b d e g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b d e h x \sqrt {- c^{2} x^{2} + 1}}{16 c^{3}} + \frac {8 b d e i x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {2 b e^{2} f \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b e^{2} g x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b e^{2} h x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {5 b e^{2} i x^{3} \sqrt {- c^{2} x^{2} + 1}}{144 c^{3}} - \frac {3 b d^{2} i \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {3 b d e h \operatorname {asin}{\left (c x \right )}}{16 c^{4}} - \frac {3 b e^{2} g \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {16 b d e i \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {8 b e^{2} h \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {5 b e^{2} i x \sqrt {- c^{2} x^{2} + 1}}{96 c^{5}} - \frac {5 b e^{2} i \operatorname {asin}{\left (c x \right )}}{96 c^{6}} & \text {for}\: c \neq 0 \\a \left (d^{2} f x + \frac {d^{2} g x^{2}}{2} + \frac {d^{2} h x^{3}}{3} + \frac {d^{2} i x^{4}}{4} + d e f x^{2} + \frac {2 d e g x^{3}}{3} + \frac {d e h x^{4}}{2} + \frac {2 d e i x^{5}}{5} + \frac {e^{2} f x^{3}}{3} + \frac {e^{2} g x^{4}}{4} + \frac {e^{2} h x^{5}}{5} + \frac {e^{2} i x^{6}}{6}\right ) & \text {otherwise} \end {cases} \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. 1287 vs.
\(2 (449) = 898\).
time = 0.44, size = 1287, normalized size = 2.66 \begin {gather*} \frac {1}{6} \, a e^{2} i x^{6} + \frac {1}{5} \, a e^{2} h x^{5} + \frac {2}{5} \, a d e i x^{5} + \frac {1}{4} \, a e^{2} g x^{4} + \frac {1}{2} \, a d e h x^{4} + \frac {1}{4} \, a d^{2} i x^{4} + \frac {1}{3} \, a e^{2} f x^{3} + \frac {2}{3} \, a d e g x^{3} + \frac {1}{3} \, a d^{2} h x^{3} + b d^{2} f x \arcsin \left (c x\right ) + a d^{2} f x + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} f x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b d e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d e f x}{2 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} g x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d e f \arcsin \left (c x\right )}{c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b e^{2} f x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, b d e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b d^{2} h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} h x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d e i x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} f}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} g x}{16 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e h x}{8 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} i x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d e f}{c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{2} g}{2 \, c^{2}} + \frac {b d e f \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b d^{2} g \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} g \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d e h \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} i \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b e^{2} h x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b d e i x \arcsin \left (c x\right )}{5 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} f}{9 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e g}{9 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} h}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} g x}{32 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d e h x}{16 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} i x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2} i x}{36 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} g \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d e h \arcsin \left (c x\right )}{c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} i \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b e^{2} i \arcsin \left (c x\right )}{6 \, c^{6}} + \frac {b e^{2} h x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, b d e i x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2} f}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e g}{3 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} h}{3 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2} h}{25 \, c^{5}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e i}{25 \, c^{5}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} i x}{144 \, c^{5}} + \frac {5 \, b e^{2} g \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {5 \, b d e h \arcsin \left (c x\right )}{16 \, c^{4}} + \frac {5 \, b d^{2} i \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} i \arcsin \left (c x\right )}{2 \, c^{6}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} h}{15 \, c^{5}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e i}{15 \, c^{5}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} i x}{96 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} i \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2} h}{5 \, c^{5}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e i}{5 \, c^{5}} + \frac {11 \, b e^{2} i \arcsin \left (c x\right )}{96 \, c^{6}} \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 \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2\,\left (i\,x^3+h\,x^2+g\,x+f\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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