Optimal. Leaf size=94 \[ -\frac{4 (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac{m}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right ) \sqrt{4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}}} \]
[Out]
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Rubi [A] time = 0.290762, antiderivative size = 109, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{2 \sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 36.1067, size = 109, normalized size = 1.16 \[ \frac{8 c \left (b d + 2 c d x\right )^{m + 1} \sqrt{a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{d \left (m + 1\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**m/(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [B] time = 0.446417, size = 193, normalized size = 2.05 \[ \frac{(b+2 c x) \left (2 c x \sqrt{b^2-4 a c}+b \sqrt{b^2-4 a c}-4 a c+b^2\right ) \left (2 c x \sqrt{b^2-4 a c}+b \sqrt{b^2-4 a c}+4 a c-b^2\right ) (d (b+2 c x))^m \, _2F_1\left (\frac{3}{2},\frac{m}{2}+\frac{1}{2};\frac{m}{2}+\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{(m+1) \left (b^2-4 a c\right )^3 \sqrt{a+x (b+c x)} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^(3/2),x]
[Out]
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Maple [F] time = 0.124, size = 0, normalized size = 0. \[ \int{ \left ( 2\,cdx+bd \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^m/(c*x^2+b*x+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d \left (b + 2 c x\right )\right )^{m}}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**m/(c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")
[Out]