Optimal. Leaf size=621 \[ \frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (d e-3 c f) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{c^{3/2} \sqrt{e+f x^2} (b c-a d)^2 \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{d x \sqrt{c+d x^2} (b c-a d)}{b^2 \sqrt{e+f x^2}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}+\frac{2 d \sqrt{e} \sqrt{c+d x^2} (d e-2 c f) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{2 d x \sqrt{c+d x^2} (d e-2 c f)}{3 b f \sqrt{e+f x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.462658, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {545, 416, 531, 418, 492, 411, 422, 539} \[ \frac{c^{3/2} \sqrt{e+f x^2} (b c-a d)^2 \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{d x \sqrt{c+d x^2} (b c-a d)}{b^2 \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}-\frac{d \sqrt{e} \sqrt{c+d x^2} (d e-3 c f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{2 d \sqrt{e} \sqrt{c+d x^2} (d e-2 c f) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{2 d x \sqrt{c+d x^2} (d e-2 c f)}{3 b f \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 545
Rule 416
Rule 531
Rule 418
Rule 492
Rule 411
Rule 422
Rule 539
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx &=\frac{d \int \frac{\left (c+d x^2\right )^{3/2}}{\sqrt{e+f x^2}} \, dx}{b}+\frac{(b c-a d) \int \frac{\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx}{b}\\ &=\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}+\frac{(d (b c-a d)) \int \frac{\sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx}{b^2}+\frac{(b c-a d)^2 \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx}{b^2}+\frac{d \int \frac{-c (d e-3 c f)-2 d (d e-2 c f) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b f}\\ &=\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}+\frac{c^{3/2} (b c-a d)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{(c d (b c-a d)) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{b^2}+\frac{\left (d^2 (b c-a d)\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{b^2}-\frac{(c d (d e-3 c f)) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b f}-\frac{\left (2 d^2 (d e-2 c f)\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b f}\\ &=\frac{d (b c-a d) x \sqrt{c+d x^2}}{b^2 \sqrt{e+f x^2}}-\frac{2 d (d e-2 c f) x \sqrt{c+d x^2}}{3 b f \sqrt{e+f x^2}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}+\frac{d (b c-a d) \sqrt{e} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{d \sqrt{e} (d e-3 c f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{c^{3/2} (b c-a d)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{(d (b c-a d) e) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{b^2}+\frac{(2 d e (d e-2 c f)) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 b f}\\ &=\frac{d (b c-a d) x \sqrt{c+d x^2}}{b^2 \sqrt{e+f x^2}}-\frac{2 d (d e-2 c f) x \sqrt{c+d x^2}}{3 b f \sqrt{e+f x^2}}+\frac{d^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b f}-\frac{d (b c-a d) \sqrt{e} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{2 d \sqrt{e} (d e-2 c f) \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{d (b c-a d) \sqrt{e} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b^2 \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{d \sqrt{e} (d e-3 c f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b f^{3/2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{c^{3/2} (b c-a d)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^2 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}\\ \end{align*}
Mathematica [C] time = 1.38856, size = 350, normalized size = 0.56 \[ \frac{-i a d \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (3 a^2 d^2 f^2+3 a b d f (d e-3 c f)+b^2 \left (9 c^2 f^2-8 c d e f+2 d^2 e^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+f \left (a b^2 c d x \left (\frac{d}{c}\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )-3 i f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^3 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )-i a b d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+7 b c f-2 b d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 a b^3 f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 988, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )} \sqrt{f x^{2} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x^{2}\right )^{\frac{5}{2}}}{\left (a + b x^{2}\right ) \sqrt{e + f x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )} \sqrt{f x^{2} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]