Optimal. Leaf size=228 \[ \frac{80 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{a+b x+c x^2}}+\frac{12 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.52227, antiderivative size = 228, 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 \[ \frac{80 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{a+b x+c x^2}}+\frac{12 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 114.377, size = 219, normalized size = 0.96 \[ \frac{80 c^{2} \sqrt{a + b x + c x^{2}}}{d \left (- 4 a c + b^{2}\right )^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{12 c}{d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} + \frac{40 c \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} \sqrt{a + b x + c x^{2}}} - \frac{2}{3 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.6153, size = 200, normalized size = 0.88 \[ \frac{2 \left ((b+2 c x)^3 (a+x (b+c x)) \left (\frac{4 a c-b^2}{(a+x (b+c x))^2}+\frac{22 c}{a+x (b+c x)}+\frac{32 c^2}{(b+2 c x)^2}\right )+\frac{60 i c (b+2 c x)^{7/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}\right )}{3 \left (b^2-4 a c\right )^3 \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.055, size = 797, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (4 \, c^{4} d^{2} x^{6} + 12 \, b c^{3} d^{2} x^{5} +{\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{4} + a^{2} b^{2} d^{2} + 2 \,{\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{3} +{\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{2} + 2 \,{\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="giac")
[Out]