Optimal. Leaf size=484 \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+\frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 (2 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 i x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b e x^4 \sqrt {1-c^2 x^2} (2 d i+e h)}{25 c}+\frac {b e^2 i x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e (2 d i+e h)\right )}{225 c^3}+\frac {b x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+5 e^2 i\right )}{144 c^3}-\frac {b \sin ^{-1}(c x) \left (24 c^4 d (d g+2 e f)+9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+5 e^2 i\right )}{96 c^6}+\frac {b \sqrt {1-c^2 x^2} \left (75 x \left (24 c^4 d (d g+2 e f)+9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+5 e^2 i\right )+32 \left (225 c^4 d^2 f+50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+24 e (2 d i+e h)\right )\right )}{7200 c^5} \]
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Rubi [A] time = 2.55, 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 = {4749, 12, 1809, 780, 216} \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+\frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 (2 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 i x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e (2 d i+e h)\right )}{225 c^3}+\frac {b \sqrt {1-c^2 x^2} \left (75 x \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )+32 \left (50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+225 c^4 d^2 f+24 e (2 d i+e h)\right )\right )}{7200 c^5}-\frac {b \sin ^{-1}(c x) \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )}{96 c^6}+\frac {b x^3 \sqrt {1-c^2 x^2} \left (e^2 \left (\frac {5 i}{c^2}+9 g\right )+9 d^2 i+18 d e h\right )}{144 c}+\frac {b e x^4 \sqrt {1-c^2 x^2} (2 d i+e h)}{25 c}+\frac {b e^2 i x^5 \sqrt {1-c^2 x^2}}{36 c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 780
Rule 1809
Rule 4749
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 = 0.94, size = 380, normalized size = 0.79 \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+\frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 (2 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 i x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \left (c \sqrt {1-c^2 x^2} \left (2 c^4 \left (25 d^2 (144 f+x (36 g+x (16 h+9 i x)))+2 d e x (900 f+x (400 g+9 x (25 h+16 i x)))+e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))\right )+c^2 \left (25 d^2 (64 h+27 i x)+2 d e \left (1600 g+675 h x+384 i x^2\right )+e^2 \left (1600 f+x \left (675 g+384 h x+250 i x^2\right )\right )\right )+3 e (512 d i+256 e h+125 e i x)\right )-75 \sin ^{-1}(c x) \left (24 c^4 d (d g+2 e f)+9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+5 e^2 i\right )\right )}{7200 c^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.47, size = 621, normalized size = 1.28 \[ \frac {1200 \, a c^{6} e^{2} i x^{6} + 7200 \, a c^{6} d^{2} f x + 1440 \, {\left (a c^{6} e^{2} h + 2 \, a c^{6} d e i\right )} x^{5} + 1800 \, {\left (a c^{6} e^{2} g + 2 \, a c^{6} d e h + a c^{6} d^{2} i\right )} x^{4} + 2400 \, {\left (a c^{6} e^{2} f + 2 \, a c^{6} d e g + a c^{6} d^{2} h\right )} x^{3} + 3600 \, {\left (2 \, a c^{6} d e f + a c^{6} d^{2} g\right )} x^{2} + 15 \, {\left (80 \, b c^{6} e^{2} i x^{6} + 480 \, b c^{6} d^{2} f x - 240 \, b c^{4} d e f - 90 \, b c^{2} d e h + 96 \, {\left (b c^{6} e^{2} h + 2 \, b c^{6} d e i\right )} x^{5} + 120 \, {\left (b c^{6} e^{2} g + 2 \, b c^{6} d e h + b c^{6} d^{2} i\right )} x^{4} + 160 \, {\left (b c^{6} e^{2} f + 2 \, b c^{6} d e g + b c^{6} d^{2} h\right )} x^{3} + 240 \, {\left (2 \, b c^{6} d e f + b c^{6} d^{2} g\right )} x^{2} - 15 \, {\left (8 \, b c^{4} d^{2} + 3 \, b c^{2} e^{2}\right )} g - 5 \, {\left (9 \, b c^{2} d^{2} + 5 \, b e^{2}\right )} i\right )} \arcsin \left (c x\right ) + {\left (200 \, b c^{5} e^{2} i x^{5} + 3200 \, b c^{3} d e g + 1536 \, b c d e i + 288 \, {\left (b c^{5} e^{2} h + 2 \, b c^{5} d e i\right )} x^{4} + 50 \, {\left (9 \, b c^{5} e^{2} g + 18 \, b c^{5} d e h + {\left (9 \, b c^{5} d^{2} + 5 \, b c^{3} e^{2}\right )} i\right )} x^{3} + 32 \, {\left (25 \, b c^{5} e^{2} f + 50 \, b c^{5} d e g + 24 \, b c^{3} d e i + {\left (25 \, b c^{5} d^{2} + 12 \, b c^{3} e^{2}\right )} h\right )} x^{2} + 800 \, {\left (9 \, b c^{5} d^{2} + 2 \, b c^{3} e^{2}\right )} f + 64 \, {\left (25 \, b c^{3} d^{2} + 12 \, b c e^{2}\right )} h + 75 \, {\left (48 \, b c^{5} d e f + 18 \, b c^{3} d e h + 3 \, {\left (8 \, b c^{5} d^{2} + 3 \, b c^{3} e^{2}\right )} g + {\left (9 \, b c^{3} d^{2} + 5 \, b c e^{2}\right )} i\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{7200 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 1283, normalized size = 2.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 674, normalized size = 1.39 \[ \frac {\frac {a \left (\frac {e^{2} i \,c^{6} x^{6}}{6}+\frac {\left (2 d c e i +e^{2} c h \right ) c^{5} x^{5}}{5}+\frac {\left (c^{2} d^{2} i +2 d \,c^{2} e h +e^{2} c^{2} g \right ) c^{4} x^{4}}{4}+\frac {\left (c^{3} d^{2} h +2 d \,c^{3} e g +e^{2} f \,c^{3}\right ) c^{3} x^{3}}{3}+\frac {\left (c^{4} d^{2} g +2 d \,c^{4} e f \right ) c^{2} x^{2}}{2}+c^{6} d^{2} 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} x^{5} d e i}{5}+\frac {\arcsin \left (c x \right ) c^{6} x^{5} e^{2} h}{5}+\frac {\arcsin \left (c x \right ) c^{6} x^{4} d^{2} i}{4}+\frac {\arcsin \left (c x \right ) c^{6} x^{4} d e h}{2}+\frac {\arcsin \left (c x \right ) c^{6} x^{4} e^{2} g}{4}+\frac {\arcsin \left (c x \right ) c^{6} x^{3} d^{2} h}{3}+\frac {2 \arcsin \left (c x \right ) c^{6} x^{3} d e g}{3}+\frac {\arcsin \left (c x \right ) c^{6} x^{3} e^{2} f}{3}+\frac {\arcsin \left (c x \right ) c^{6} x^{2} d^{2} g}{2}+\arcsin \left (c x \right ) c^{6} x^{2} d e f +\arcsin \left (c x \right ) c^{6} d^{2} 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 d c e i +12 e^{2} c 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 d \,c^{2} e h +15 e^{2} c^{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 d \,c^{3} e g +20 e^{2} f \,c^{3}\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 d \,c^{4} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{60}+c^{5} d^{2} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{5}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 840, normalized size = 1.74 \[ \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} + a d e f x^{2} + \frac {1}{2} \, a d^{2} g x^{2} + \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 e f + \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 e^{2} f + \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 {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 e g + \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 e^{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 + \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 e h + \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 e^{2} h + \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 d^{2} i + \frac {2}{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 d e i + \frac {1}{288} \, {\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} i + a d^{2} f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2} f}{c} \]
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 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2\,\left (i\,x^3+h\,x^2+g\,x+f\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.85, size = 1197, normalized size = 2.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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