Optimal. Leaf size=173 \[ -\frac{8 c d^{3/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}-\frac{4 c d \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.418829, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{8 c d^{3/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}-\frac{4 c d \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 91.8256, size = 170, normalized size = 0.98 \[ - \frac{8 c d^{\frac{3}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{3 \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \sqrt{a + b x + c x^{2}}} - \frac{4 c d \sqrt{b d + 2 c d x}}{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} - \frac{2 d \sqrt{b d + 2 c d x}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a)**(5/2),x)
[Out]
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Mathematica [C] time = 0.758028, size = 168, normalized size = 0.97 \[ -\frac{2 d \sqrt{d (b+2 c x)} \left (\sqrt{b^2-4 a c} \left (2 c \left (c x^2-a\right )+b^2+2 b c x\right )-4 i \sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{5/2} \left (\frac{c (a+x (b+c x))}{(b+2 c x)^2}\right )^{3/2} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )\right )}{3 \left (b^2-4 a c\right )^{3/2} (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.069, size = 487, normalized size = 2.8 \[{\frac{2\,d}{ \left ( 12\,ac-3\,{b}^{2} \right ) \left ( 2\,cx+b \right ) } \left ( 2\,{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){x}^{2}{c}^{2}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}+2\,{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) xbc\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}+2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}ac+4\,{c}^{3}{x}^{3}+6\,b{c}^{2}{x}^{2}-4\,a{c}^{2}x+4\,x{b}^{2}c-2\,abc+{b}^{3} \right ) \sqrt{d \left ( 2\,cx+b \right ) } \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(3/2)/(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. \[{\rm integral}\left (\frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(3/2)/(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)**(3/2)/(c*x**2+b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")
[Out]