Optimal. Leaf size=258 \[ -\frac{56 c d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}+\frac{56 c d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}-\frac{28 c d^3 (b d+2 c d x)^{3/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.802349, antiderivative size = 258, 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 \[ -\frac{56 c d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}+\frac{56 c d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}-\frac{28 c d^3 (b d+2 c d x)^{3/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 142.314, size = 253, normalized size = 0.98 \[ \frac{56 c d^{\frac{9}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt{a + b x + c x^{2}}} - \frac{56 c d^{\frac{9}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{\sqrt{a + b x + c x^{2}}} - \frac{28 c d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 d \left (b d + 2 c d x\right )^{\frac{7}{2}}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]
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)**(5/2),x)
[Out]
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Mathematica [C] time = 1.64859, size = 194, normalized size = 0.75 \[ \frac{2 (d (b+2 c x))^{9/2} \left (-\frac{2 c \left (7 a+9 c x^2\right )+b^2+18 b c x}{a+x (b+c x)}+\frac{84 i c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}\right )}{3 (b+2 c x)^3 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.041, size = 859, normalized size = 3.3 \[ \text{result too large to display} \]
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)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]
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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} \sqrt{2 \, c d x + b d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")
[Out]
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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)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]
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)^(5/2),x, algorithm="giac")
[Out]