Optimal. Leaf size=326 \[ -\frac{d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{195 c^2}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{78 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{13 c d} \]
[Out]
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Rubi [A] time = 0.989742, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{195 c^2}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{78 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{13 c d} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 165.259, size = 308, normalized size = 0.94 \[ \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{13 c d} + \frac{d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{195 c^{2}} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}}{78 c^{2} d} + \frac{d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{15}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{130 c^{3} \sqrt{a + b x + c x^{2}}} - \frac{d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{15}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{130 c^{3} \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(5/2)*(c*x**2+b*x+a)**(3/2),x)
[Out]
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Mathematica [C] time = 1.83277, size = 256, normalized size = 0.79 \[ \frac{(d (b+2 c x))^{5/2} \left (\frac{c (a+x (b+c x)) \left (8 c^2 \left (4 a^2+25 a c x^2+15 c^2 x^4\right )+2 b^2 c \left (17 a+65 c x^2\right )+40 b c^2 x \left (5 a+6 c x^2\right )-3 b^4+10 b^3 c x\right )}{b+2 c x}-\frac{3 i \left (b^2-4 a c\right )^{5/2} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{5/2}}\right )}{390 c^3 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.032, size = 938, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(5/2)*(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. \[{\rm integral}\left ({\left (4 \, c^{3} d^{2} x^{4} + 8 \, b c^{2} d^{2} x^{3} + a b^{2} d^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{2} +{\left (b^{3} + 4 \, a b c\right )} d^{2} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2),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)**(5/2)*(c*x**2+b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 1.40137, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")
[Out]