Optimal. Leaf size=247 \[ \frac{1040}{7} c^2 d^7 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{520 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{7 \sqrt{a+b x+c x^2}}+\frac{624}{7} c^2 d^5 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.607228, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{1040}{7} c^2 d^7 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{520 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{7 \sqrt{a+b x+c x^2}}+\frac{624}{7} c^2 d^5 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{13/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 125.344, size = 245, normalized size = 0.99 \[ \frac{1040 c^{2} d^{7} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{7} + \frac{624 c^{2} d^{5} \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{7} + \frac{520 c d^{\frac{15}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{7 \sqrt{a + b x + c x^{2}}} - \frac{52 c d^{3} \left (b d + 2 c d x\right )^{\frac{9}{2}}}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 d \left (b d + 2 c d x\right )^{\frac{13}{2}}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(15/2)/(c*x**2+b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 2.67516, size = 231, normalized size = 0.94 \[ \frac{(d (b+2 c x))^{15/2} \left (\frac{2 (a+x (b+c x)) \left (-64 c^2 \left (38 a c-11 b^2\right )-\frac{266 c \left (b^2-4 a c\right )^2}{a+x (b+c x)}-\frac{7 \left (b^2-4 a c\right )^3}{(a+x (b+c x))^2}+384 b c^3 x+384 c^4 x^2\right )}{3 (b+2 c x)^7}+\frac{520 i c \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{13/2}}\right )}{7 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.087, size = 1473, normalized size = 6. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(15/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (128 \, c^{7} d^{7} x^{7} + 448 \, b c^{6} d^{7} x^{6} + 672 \, b^{2} c^{5} d^{7} x^{5} + 560 \, b^{3} c^{4} d^{7} x^{4} + 280 \, b^{4} c^{3} d^{7} x^{3} + 84 \, b^{5} c^{2} d^{7} x^{2} + 14 \, b^{6} c d^{7} x + b^{7} d^{7}\right )} \sqrt{2 \, c d x + b d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^(15/2)/(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((2*c*d*x+b*d)**(15/2)/(c*x**2+b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.657267, 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)^(15/2)/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")
[Out]