Optimal. Leaf size=273 \[ -\frac{14 d^{9/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c \sqrt{a+b x+c x^2}}+\frac{14 d^{9/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c \sqrt{a+b x+c x^2}}+\frac{28}{45} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}+\frac{4}{9} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2} \]
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Rubi [A] time = 0.850666, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{14 d^{9/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c \sqrt{a+b x+c x^2}}+\frac{14 d^{9/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c \sqrt{a+b x+c x^2}}+\frac{28}{45} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}+\frac{4}{9} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(9/2)/Sqrt[a + b*x + c*x^2],x]
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Rubi in Sympy [A] time = 147.205, size = 264, normalized size = 0.97 \[ \frac{28 d^{3} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{45} + \frac{4 d \left (b d + 2 c d x\right )^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}}{9} + \frac{14 d^{\frac{9}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{15 c \sqrt{a + b x + c x^{2}}} - \frac{14 d^{\frac{9}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{15 c \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(1/2),x)
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Mathematica [C] time = 1.40713, size = 205, normalized size = 0.75 \[ \frac{2 (d (b+2 c x))^{9/2} \left (8 (a+x (b+c x)) \left (c \left (5 c x^2-7 a\right )+3 b^2+5 b c x\right )+\frac{21 i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{c \left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}\right )}{45 (b+2 c x)^3 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(9/2)/Sqrt[a + b*x + c*x^2],x]
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Maple [B] time = 0.047, size = 703, normalized size = 2.6 \[{\frac{{d}^{4}}{45\,c \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 320\,{c}^{6}{x}^{6}+960\,b{c}^{5}{x}^{5}+1344\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{3}{c}^{3}-1008\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{b}^{2}{c}^{2}+252\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{4}c-21\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{6}-128\,{x}^{4}a{c}^{5}+1232\,{x}^{4}{b}^{2}{c}^{4}-256\,{x}^{3}ab{c}^{4}+864\,{b}^{3}{c}^{3}{x}^{3}-448\,{x}^{2}{a}^{2}{c}^{4}+32\,{x}^{2}a{b}^{2}{c}^{3}+320\,{x}^{2}{b}^{4}{c}^{2}-448\,{a}^{2}b{c}^{3}x+160\,a{b}^{3}{c}^{2}x+48\,{b}^{5}cx-112\,{a}^{2}{b}^{2}{c}^{2}+48\,a{b}^{4}c \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} \sqrt{2 \, c d x + b d}}{\sqrt{c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a),x, algorithm="giac")
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