Optimal. Leaf size=310 \[ -\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{12 c^3 d^5 \sqrt{b d+2 c d x}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}} \]
[Out]
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Rubi [A] time = 0.917426, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{12 c^3 d^5 \sqrt{b d+2 c d x}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 170.035, size = 292, normalized size = 0.94 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{9 c d \left (b d + 2 c d x\right )^{\frac{9}{2}}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{18 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{\sqrt{a + b x + c x^{2}}}{12 c^{3} d^{5} \sqrt{b d + 2 c d x}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{12 c^{4} d^{\frac{11}{2}} \sqrt{a + b x + c x^{2}}} - \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{12 c^{4} d^{\frac{11}{2}} \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(11/2),x)
[Out]
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Mathematica [C] time = 1.35914, size = 251, normalized size = 0.81 \[ \frac{-\frac{(b+2 c x) (a+x (b+c x)) \left (4 c^2 \left (a^2+4 a c x^2+15 c^2 x^4\right )+2 b^2 c \left (a+43 c x^2\right )+8 b c^2 x \left (2 a+15 c x^2\right )+3 b^4+26 b^3 c x\right )}{3 c^3}+\frac{i (b+2 c x)^7 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{c^4 \left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}}{12 \sqrt{a+x (b+c x)} (d (b+2 c x))^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(11/2),x]
[Out]
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Maple [B] time = 0.043, size = 1489, normalized size = 4.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(11/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(11/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{{\left (32 \, c^{5} d^{5} x^{5} + 80 \, b c^{4} d^{5} x^{4} + 80 \, b^{2} c^{3} d^{5} x^{3} + 40 \, b^{3} c^{2} d^{5} x^{2} + 10 \, b^{4} c d^{5} x + b^{5} d^{5}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(11/2),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((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(11/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(11/2),x, algorithm="giac")
[Out]