Optimal. Leaf size=184 \[ -\frac{20 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}-\frac{40 c \sqrt{a+b x+c x^2}}{3 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.429715, antiderivative size = 184, 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 \[ -\frac{20 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}-\frac{40 c \sqrt{a+b x+c x^2}}{3 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 101.723, size = 177, normalized size = 0.96 \[ - \frac{40 c \sqrt{a + b x + c x^{2}}}{3 d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}} - \frac{2}{d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} - \frac{20 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{3 d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} \]
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)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.66817, size = 177, normalized size = 0.96 \[ -\frac{2 \left (\sqrt{-\sqrt{b^2-4 a c}} \left (4 c \left (2 a+5 c x^2\right )+3 b^2+20 b c x\right )+10 i (b+2 c x)^{5/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )\right )}{3 d \sqrt{-\sqrt{b^2-4 a c}} \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.046, size = 365, normalized size = 2. \[ -{\frac{2}{3\,{d}^{3} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( 2\,cx+b \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 10\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}xc+5\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}b+20\,{c}^{2}{x}^{2}+20\,bxc+8\,ac+3\,{b}^{2} \right ) } \]
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)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]
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)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (4 \, c^{3} d^{2} x^{4} + 8 \, b c^{2} d^{2} x^{3} + a b^{2} d^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{2} +{\left (b^{3} + 4 \, a b c\right )} d^{2} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]
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)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]
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)^(3/2)),x, algorithm="giac")
[Out]