Optimal. Leaf size=484 \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+\frac{1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 (2 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^2 i x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e (2 d i+e h)\right )}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left (75 x \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )+32 \left (50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+225 c^4 d^2 f+24 e (2 d i+e h)\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )}{96 c^6}+\frac{b x^3 \sqrt{1-c^2 x^2} \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+5 e^2 i\right )}{144 c^3}+\frac{b e x^4 \sqrt{1-c^2 x^2} (2 d i+e h)}{25 c}+\frac{b e^2 i x^5 \sqrt{1-c^2 x^2}}{36 c} \]
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Rubi [A] time = 2.54995, antiderivative size = 482, normalized size of antiderivative = 1., 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 4749
Rule 12
Rule 1809
Rule 780
Rule 216
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*}
Mathematica [A] time = 0.905933, size = 380, normalized size = 0.79 \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+\frac{1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 (2 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^2 i x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \left (c \sqrt{1-c^2 x^2} \left (2 c^4 \left (25 d^2 (144 f+x (36 g+x (16 h+9 i x)))+2 d e x (900 f+x (400 g+9 x (25 h+16 i x)))+e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))\right )+c^2 \left (25 d^2 (64 h+27 i x)+2 d e \left (1600 g+675 h x+384 i x^2\right )+e^2 \left (1600 f+x \left (675 g+384 h x+250 i x^2\right )\right )\right )+3 e (512 d i+256 e h+125 e i x)\right )-75 \sin ^{-1}(c x) \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )\right )}{7200 c^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 674, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54514, size = 1231, normalized size = 2.54 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.46506, size = 1426, normalized size = 2.95 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.5868, size = 1197, normalized size = 2.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.42659, size = 1917, normalized size = 3.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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