Optimal. Leaf size=219 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{84 c^4 d^{5/2} \sqrt{a+b x+c x^2}}-\frac{5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{84 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}} \]
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Rubi [A] time = 0.518658, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{84 c^4 d^{5/2} \sqrt{a+b x+c x^2}}-\frac{5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{84 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 111.101, size = 211, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{3 c d \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{5 \sqrt{b d + 2 c d x} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{42 c^{2} d^{3}} - \frac{5 \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{84 c^{3} d^{3}} + \frac{5 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{84 c^{4} d^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}} \]
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)**(5/2),x)
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Mathematica [C] time = 1.23309, size = 201, normalized size = 0.92 \[ \frac{\frac{(b+2 c x)^3 (a+x (b+c x)) \left (-\frac{7 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}+64 a c-13 b^2+12 b c x+12 c^2 x^2\right )}{4 c^3}+\frac{5 i \left (b^2-4 a c\right )^2 (b+2 c x)^{7/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{c^4 \sqrt{-\sqrt{b^2-4 a c}}}}{84 \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(5/2),x]
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Maple [B] time = 0.042, size = 1000, normalized size = 4.6 \[ \text{result too large to display} \]
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)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}\,{d x} \]
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)^(5/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{{\left (4 \, c^{2} d^{2} x^{2} + 4 \, b c d^{2} x + b^{2} d^{2}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]
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)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}}}\, dx \]
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)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}\,{d x} \]
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)^(5/2),x, algorithm="giac")
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