Optimal. Leaf size=304 \[ -\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{2 d \sqrt{x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{2 d^2 x^{5/2} (3 b c-a d)}{5 b^2}+\frac{2 d^3 x^{9/2}}{9 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.506925, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{2 d \sqrt{x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{2 d^2 x^{5/2} (3 b c-a d)}{5 b^2}+\frac{2 d^3 x^{9/2}}{9 b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^3/(Sqrt[x]*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 96.7473, size = 289, normalized size = 0.95 \[ \frac{2 d^{3} x^{\frac{9}{2}}}{9 b} - \frac{2 d^{2} x^{\frac{5}{2}} \left (a d - 3 b c\right )}{5 b^{2}} + \frac{2 d \sqrt{x} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{b^{3}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{3}{4}} b^{\frac{13}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{3}{4}} b^{\frac{13}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{13}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{13}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**3/(b*x**2+a)/x**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.260934, size = 291, normalized size = 0.96 \[ \frac{-72 a^{3/4} b^{5/4} d^2 x^{5/2} (a d-3 b c)+40 a^{3/4} b^{9/4} d^3 x^{9/2}+360 a^{3/4} \sqrt [4]{b} d \sqrt{x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )-45 \sqrt{2} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+45 \sqrt{2} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-90 \sqrt{2} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+90 \sqrt{2} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{180 a^{3/4} b^{13/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^3/(Sqrt[x]*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.014, size = 650, normalized size = 2.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^3/(b*x^2+a)/x^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/((b*x^2 + a)*sqrt(x)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.254648, size = 2052, normalized size = 6.75 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/((b*x^2 + a)*sqrt(x)),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((d*x**2+c)**3/(b*x**2+a)/x**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.284875, size = 662, normalized size = 2.18 \[ \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a b^{4}} + \frac{2 \,{\left (5 \, b^{8} d^{3} x^{\frac{9}{2}} + 27 \, b^{8} c d^{2} x^{\frac{5}{2}} - 9 \, a b^{7} d^{3} x^{\frac{5}{2}} + 135 \, b^{8} c^{2} d \sqrt{x} - 135 \, a b^{7} c d^{2} \sqrt{x} + 45 \, a^{2} b^{6} d^{3} \sqrt{x}\right )}}{45 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/((b*x^2 + a)*sqrt(x)),x, algorithm="giac")
[Out]