Optimal. Leaf size=439 \[ -\frac{13 a^2 d^2-10 a b c d+5 b^2 c^2}{20 c^2 d \sqrt{x} \left (c+d x^2\right )^2}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt{x}}-\frac{5 b^2 c^2-9 a d (10 b c-13 a d)}{80 c^3 d \sqrt{x} \left (c+d x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.979314, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ -\frac{13 a^2 d^2-10 a b c d+5 b^2 c^2}{20 c^2 d \sqrt{x} \left (c+d x^2\right )^2}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac{\frac{5 b^2}{d}-\frac{9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt{x} \left (c+d x^2\right )}+\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} d^{3/4}}-\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} d^{3/4}}+\frac{5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 117.404, size = 420, normalized size = 0.96 \[ - \frac{2 a^{2}}{5 c x^{\frac{5}{2}} \left (c + d x^{2}\right )^{2}} - \frac{a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}}{20 c^{2} d \sqrt{x} \left (c + d x^{2}\right )^{2}} - \frac{9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}}{80 c^{3} d \sqrt{x} \left (c + d x^{2}\right )} + \frac{9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}}{16 c^{4} d \sqrt{x}} + \frac{\sqrt{2} \left (9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{17}{4}} d^{\frac{3}{4}}} - \frac{\sqrt{2} \left (9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{128 c^{\frac{17}{4}} d^{\frac{3}{4}}} - \frac{\sqrt{2} \left (9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{17}{4}} d^{\frac{3}{4}}} + \frac{\sqrt{2} \left (9 a d \left (13 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{64 c^{\frac{17}{4}} d^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**(7/2)/(d*x**2+c)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.767862, size = 382, normalized size = 0.87 \[ \frac{\frac{40 \sqrt [4]{c} x^{3/2} \left (21 a^2 d^2-26 a b c d+5 b^2 c^2\right )}{c+d x^2}+\frac{5 \sqrt{2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{3/4}}-\frac{5 \sqrt{2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{3/4}}-\frac{10 \sqrt{2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{3/4}}+\frac{10 \sqrt{2} \left (117 a^2 d^2-90 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{3/4}}-\frac{256 a^2 c^{5/4}}{x^{5/2}}+\frac{160 c^{5/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{1280 a \sqrt [4]{c} (3 a d-2 b c)}{\sqrt{x}}}{640 c^{17/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^(7/2)*(c + d*x^2)^3),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.034, size = 590, normalized size = 1.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^(7/2)/(d*x^2+c)^3,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((b*x^2 + a)^2/((d*x^2 + c)^3*x^(7/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.280623, size = 2168, normalized size = 4.94 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^3*x^(7/2)),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((b*x**2+a)**2/x**(7/2)/(d*x**2+c)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.31651, size = 599, normalized size = 1.36 \[ \frac{5 \, b^{2} c^{2} d x^{\frac{7}{2}} - 26 \, a b c d^{2} x^{\frac{7}{2}} + 21 \, a^{2} d^{3} x^{\frac{7}{2}} + 9 \, b^{2} c^{3} x^{\frac{3}{2}} - 34 \, a b c^{2} d x^{\frac{3}{2}} + 25 \, a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c^{4}} - \frac{2 \,{\left (10 \, a b c x^{2} - 15 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, c^{4} x^{\frac{5}{2}}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{5} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{5} d^{3}} - \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{5} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{5} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^3*x^(7/2)),x, algorithm="giac")
[Out]