Optimal. Leaf size=421 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c^2 e^5}+\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{4 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (a d+10 b c)}{5 c^2 e^3 \sqrt{e x}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.963847, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{5 c e (e x)^{5/2}}+\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c^2 e^5}+\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (a d (a d+10 b c)+b^2 c^2\right )}{5 c \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{4 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (a d+10 b c)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (a d+10 b c)}{5 c^2 e^3 \sqrt{e x}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(7/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 94.1369, size = 394, normalized size = 0.94 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{5 c e \left (e x\right )^{\frac{5}{2}}} - \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d + 10 b c\right )}{5 c^{2} e^{3} \sqrt{e x}} + \frac{4 \sqrt{e x} \sqrt{c + d x^{2}} \left (a d \left (a d + 10 b c\right ) + b^{2} c^{2}\right )}{5 c \sqrt{d} e^{4} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{2 \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d \left (a d + 10 b c\right ) + b^{2} c^{2}\right )}{5 c^{2} e^{5}} - \frac{4 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (a d + 10 b c\right ) + b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} + \frac{2 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (a d + 10 b c\right ) + b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/(e*x)**(7/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.40011, size = 226, normalized size = 0.54 \[ \frac{x^{7/2} \left (\frac{2 \sqrt{c+d x^2} \left (-a^2 \left (c+2 d x^2\right )-10 a b c x^2+b^2 c x^4\right )}{c x^{5/2}}-\frac{4 x \left (a^2 d^2+10 a b c d+b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )}{c d \sqrt{c+d x^2}}\right )}{5 (e x)^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/(e*x)^(7/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.074, size = 648, normalized size = 1.5 \[{\frac{2}{5\,d{x}^{2}{e}^{3}c} \left ( 2\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c{d}^{2}+20\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d+2\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}{c}^{3}-\sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}{a}^{2}c{d}^{2}-10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{2}ab{c}^{2}d-\sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}{b}^{2}{c}^{3}+{x}^{6}{b}^{2}c{d}^{2}-2\,{x}^{4}{a}^{2}{d}^{3}-10\,{x}^{4}abc{d}^{2}+{x}^{4}{b}^{2}{c}^{2}d-3\,{x}^{2}{a}^{2}c{d}^{2}-10\,{x}^{2}ab{c}^{2}d-{a}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^(1/2)/(e*x)^(7/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/(e*x)^(7/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{\sqrt{e x} e^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/(e*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*(d*x**2+c)**(1/2)/(e*x)**(7/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/(e*x)^(7/2),x, algorithm="giac")
[Out]