Optimal. Leaf size=361 \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e x^4 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b e x^3 \sqrt {1-c^2 x^2} (2 d h+e g)}{16 c}+\frac {b e^2 h x^4 \sqrt {1-c^2 x^2}}{25 c}-\frac {b \sin ^{-1}(c x) \left (8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right )}{32 c^4}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e^2 h\right )}{225 c^3}+\frac {b \sqrt {1-c^2 x^2} \left (225 c^2 x \left (8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right )+32 \left (225 c^4 d^2 f+50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+24 e^2 h\right )\right )}{7200 c^5} \]
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Rubi [A] time = 1.22, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4749, 12, 1809, 780, 216} \[ \frac {1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e x^4 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e^2 h\right )}{225 c^3}+\frac {b \sqrt {1-c^2 x^2} \left (32 \left (50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+225 c^4 d^2 f+24 e^2 h\right )+225 c^2 x \left (8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right )\right )}{7200 c^5}-\frac {b \sin ^{-1}(c x) \left (8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right )}{32 c^4}+\frac {b e x^3 \sqrt {1-c^2 x^2} (2 d h+e g)}{16 c}+\frac {b e^2 h x^4 \sqrt {1-c^2 x^2}}{25 c} \]
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\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} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (60 d^2 f+30 d (2 e f+d g) x+20 \left (e^2 f+2 d e g+d^2 h\right ) x^2+15 e (e g+2 d h) x^3+12 e^2 h x^4\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} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{60} (b c) \int \frac {x \left (60 d^2 f+30 d (2 e f+d g) x+20 \left (e^2 f+2 d e g+d^2 h\right ) x^2+15 e (e g+2 d h) x^3+12 e^2 h x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b e^2 h x^4 \sqrt {1-c^2 x^2}}{25 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} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-300 c^2 d^2 f-150 c^2 d (2 e f+d g) x-4 \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2-75 c^2 e (e g+2 d h) x^3\right )}{\sqrt {1-c^2 x^2}} \, dx}{300 c}\\ &=\frac {b e (e g+2 d h) x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b e^2 h x^4 \sqrt {1-c^2 x^2}}{25 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} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (1200 c^4 d^2 f+75 c^2 \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right ) x+16 c^2 \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{1200 c^3}\\ &=\frac {b \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e (e g+2 d h) x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b e^2 h x^4 \sqrt {1-c^2 x^2}}{25 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} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-16 c^2 \left (225 c^4 d^2 f+24 e^2 h+50 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right )-225 c^4 \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{3600 c^5}\\ &=\frac {b \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e (e g+2 d h) x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b e^2 h x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b \left (32 \left (225 c^4 d^2 f+24 e^2 h+50 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right )+225 c^2 \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\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} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac {b \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e (e g+2 d h) x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b e^2 h x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b \left (32 \left (225 c^4 d^2 f+24 e^2 h+50 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right )+225 c^2 \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}-\frac {b \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right ) \sin ^{-1}(c x)}{32 c^4}+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} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.53, size = 307, normalized size = 0.85 \[ \frac {120 a c^5 x \left (10 d^2 (6 f+x (3 g+2 h x))+10 d e x (6 f+x (4 g+3 h x))+e^2 x^2 (20 f+3 x (5 g+4 h x))\right )+b \sqrt {1-c^2 x^2} \left (2 c^4 \left (100 d^2 (36 f+x (9 g+4 h x))+50 d e x (36 f+x (16 g+9 h x))+e^2 x^2 (400 f+9 x (25 g+16 h x))\right )+c^2 \left (1600 d^2 h+50 d e (64 g+27 h x)+e^2 \left (1600 f+675 g x+384 h x^2\right )\right )+768 e^2 h\right )+15 b c \sin ^{-1}(c x) \left (8 c^4 x \left (10 d^2 (6 f+x (3 g+2 h x))+10 d e x (6 f+x (4 g+3 h x))+e^2 x^2 (20 f+3 x (5 g+4 h x))\right )-120 c^2 d (d g+2 e f)-45 e (2 d h+e g)\right )}{7200 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 449, normalized size = 1.24 \[ \frac {1440 \, a c^{5} e^{2} h x^{5} + 7200 \, a c^{5} d^{2} f x + 1800 \, {\left (a c^{5} e^{2} g + 2 \, a c^{5} d e h\right )} x^{4} + 2400 \, {\left (a c^{5} e^{2} f + 2 \, a c^{5} d e g + a c^{5} d^{2} h\right )} x^{3} + 3600 \, {\left (2 \, a c^{5} d e f + a c^{5} d^{2} g\right )} x^{2} + 15 \, {\left (96 \, b c^{5} e^{2} h x^{5} + 480 \, b c^{5} d^{2} f x - 240 \, b c^{3} d e f - 90 \, b c d e h + 120 \, {\left (b c^{5} e^{2} g + 2 \, b c^{5} d e h\right )} x^{4} + 160 \, {\left (b c^{5} e^{2} f + 2 \, b c^{5} d e g + b c^{5} d^{2} h\right )} x^{3} + 240 \, {\left (2 \, b c^{5} d e f + b c^{5} d^{2} g\right )} x^{2} - 15 \, {\left (8 \, b c^{3} d^{2} + 3 \, b c e^{2}\right )} g\right )} \arcsin \left (c x\right ) + {\left (288 \, b c^{4} e^{2} h x^{4} + 3200 \, b c^{2} d e g + 450 \, {\left (b c^{4} e^{2} g + 2 \, b c^{4} d e h\right )} x^{3} + 32 \, {\left (25 \, b c^{4} e^{2} f + 50 \, b c^{4} d e g + {\left (25 \, b c^{4} d^{2} + 12 \, b c^{2} e^{2}\right )} h\right )} x^{2} + 800 \, {\left (9 \, b c^{4} d^{2} + 2 \, b c^{2} e^{2}\right )} f + 64 \, {\left (25 \, b c^{2} d^{2} + 12 \, b e^{2}\right )} h + 225 \, {\left (16 \, b c^{4} d e f + 6 \, b c^{2} d e h + {\left (8 \, b c^{4} d^{2} + 3 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{7200 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.54, size = 844, normalized size = 2.34 \[ \frac {1}{5} \, a h x^{5} e^{2} + \frac {1}{2} \, a d h x^{4} e + \frac {1}{3} \, a d^{2} h x^{3} + \frac {1}{4} \, a g x^{4} e^{2} + \frac {2}{3} \, a d g x^{3} e + b d^{2} f x \arcsin \left (c x\right ) + \frac {1}{3} \, a f x^{3} e^{2} + a d^{2} f x + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b d g x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} g x}{4 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b d f x e}{2 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b d^{2} h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b f x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d f \arcsin \left (c x\right ) e}{c^{2}} + \frac {2 \, b d g x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} f}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d h x e}{8 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{2} g}{2 \, c^{2}} + \frac {b d^{2} g \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {b f x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b h x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d f e}{c^{2}} + \frac {b d f \arcsin \left (c x\right ) e}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d h \arcsin \left (c x\right ) e}{2 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} h}{9 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b g x e^{2}}{16 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d g e}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d h x e}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b g \arcsin \left (c x\right ) e^{2}}{4 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b h x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d h \arcsin \left (c x\right ) e}{c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} h}{3 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b f e^{2}}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b g x e^{2}}{32 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d g e}{3 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} b g \arcsin \left (c x\right ) e^{2}}{2 \, c^{4}} + \frac {b h x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {5 \, b d h \arcsin \left (c x\right ) e}{16 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b f e^{2}}{3 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b h e^{2}}{25 \, c^{5}} + \frac {5 \, b g \arcsin \left (c x\right ) e^{2}}{32 \, c^{4}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b h e^{2}}{15 \, c^{5}} + \frac {\sqrt {-c^{2} x^{2} + 1} b h e^{2}}{5 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 502, normalized size = 1.39 \[ \frac {\frac {a \left (\frac {e^{2} h \,c^{5} x^{5}}{5}+\frac {\left (2 d c e h +e^{2} c g \right ) c^{4} x^{4}}{4}+\frac {\left (c^{2} d^{2} h +2 d \,c^{2} e g +e^{2} f \,c^{2}\right ) c^{3} x^{3}}{3}+\frac {\left (c^{3} d^{2} g +2 d \,c^{3} e f \right ) c^{2} x^{2}}{2}+c^{5} d^{2} f x \right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} h \,c^{5} x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{5} x^{4} d e h}{2}+\frac {\arcsin \left (c x \right ) c^{5} x^{4} e^{2} g}{4}+\frac {\arcsin \left (c x \right ) c^{5} x^{3} d^{2} h}{3}+\frac {2 \arcsin \left (c x \right ) c^{5} x^{3} d e g}{3}+\frac {\arcsin \left (c x \right ) c^{5} x^{3} e^{2} f}{3}+\frac {\arcsin \left (c x \right ) c^{5} x^{2} d^{2} g}{2}+\arcsin \left (c x \right ) c^{5} x^{2} d e f +\arcsin \left (c x \right ) c^{5} d^{2} f x -\frac {e^{2} h \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {\left (30 d c e h +15 e^{2} c g \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{60}-\frac {\left (20 c^{2} d^{2} h +40 d \,c^{2} e g +20 e^{2} f \,c^{2}\right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{60}-\frac {\left (30 c^{3} d^{2} g +60 d \,c^{3} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{60}+c^{4} d^{2} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{4}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 581, normalized size = 1.61 \[ \frac {1}{5} \, a e^{2} h x^{5} + \frac {1}{4} \, a e^{2} g x^{4} + \frac {1}{2} \, a d e h 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 + 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 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2\,\left (h\,x^2+g\,x+f\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.78, size = 821, normalized size = 2.27 \[ \begin {cases} a d^{2} f x + \frac {a d^{2} g x^{2}}{2} + \frac {a d^{2} h x^{3}}{3} + a d e f x^{2} + \frac {2 a d e g x^{3}}{3} + \frac {a d e h x^{4}}{2} + \frac {a e^{2} f x^{3}}{3} + \frac {a e^{2} g x^{4}}{4} + \frac {a e^{2} h x^{5}}{5} + 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} + 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 {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 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 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 {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 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 {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 {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 {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 {8 b e^{2} h \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} & \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} + d e f x^{2} + \frac {2 d e g x^{3}}{3} + \frac {d e h x^{4}}{2} + \frac {e^{2} f x^{3}}{3} + \frac {e^{2} g x^{4}}{4} + \frac {e^{2} h x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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