Optimal. Leaf size=484 \[ -\frac {c (2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\sqrt {c} \left (3 a f^2+2 c \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 f^3}-\frac {\left (c e \left (e-\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )-2 f \left (2 a c d f^2-a^2 f^3+c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (c e \left (e+\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )-2 f \left (2 a c d f^2-a^2 f^3+c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.76, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {993, 1094, 223,
212, 1048, 739} \begin {gather*} \frac {\left (-2 a^2 f^4-c e \left (e-\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )+4 a c d f^3+2 c^2 d f \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\left (-2 a^2 f^4-c e \left (\sqrt {e^2-4 d f}+e\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )+4 a c d f^3+2 c^2 d f \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-d f\right )\right )}{2 f^3}-\frac {c \sqrt {a+c x^2} (2 e-f x)}{2 f^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 739
Rule 993
Rule 1048
Rule 1094
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx &=-\frac {c (2 e-f x) \sqrt {a+c x^2}}{2 f^2}-\frac {\int \frac {a f (c d-2 a f)-c e (2 c d-a f) x-c \left (3 a f^2+2 c \left (e^2-d f\right )\right ) x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 f^2}\\ &=-\frac {c (2 e-f x) \sqrt {a+c x^2}}{2 f^2}-\frac {\int \frac {a f^2 (c d-2 a f)+c d \left (3 a f^2+2 c \left (e^2-d f\right )\right )+\left (-c e f (2 c d-a f)+c e \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 f^3}+\frac {\left (c \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 f^3}\\ &=-\frac {c (2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\left (c \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 f^3}-\frac {\left (2 f \left (a f^2 (c d-2 a f)+c d \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (-c e f (2 c d-a f)+c e \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{2 f^3 \sqrt {e^2-4 d f}}+\frac {\left (2 f \left (a f^2 (c d-2 a f)+c d \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (-c e f (2 c d-a f)+c e \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{2 f^3 \sqrt {e^2-4 d f}}\\ &=-\frac {c (2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\sqrt {c} \left (3 a f^2+2 c \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 f^3}+\frac {\left (2 f \left (a f^2 (c d-2 a f)+c d \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (-c e f (2 c d-a f)+c e \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{2 f^3 \sqrt {e^2-4 d f}}-\frac {\left (2 f \left (a f^2 (c d-2 a f)+c d \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (-c e f (2 c d-a f)+c e \left (3 a f^2+2 c \left (e^2-d f\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{2 f^3 \sqrt {e^2-4 d f}}\\ &=-\frac {c (2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\sqrt {c} \left (3 a f^2+2 c \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 f^3}-\frac {\left (c e \left (e-\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )-2 f \left (2 a c d f^2-a^2 f^3+c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (c e \left (e+\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )-2 f \left (2 a c d f^2-a^2 f^3+c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.57, size = 542, normalized size = 1.12 \begin {gather*} \frac {c f (-2 e+f x) \sqrt {a+c x^2}+\sqrt {c} \left (-2 c e^2+2 c d f-3 a f^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )+2 \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {a c^2 e^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-2 a c^2 d e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 a^2 c e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 c^{5/2} d e^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 c^{5/2} d^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a c^{3/2} d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} f^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c^2 e^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 c^2 d e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 a c e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{2 f^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2260\) vs.
\(2(435)=870\).
time = 0.14, size = 2261, normalized size = 4.67
method | result | size |
default | \(\text {Expression too large to display}\) | \(2261\) |
risch | \(\text {Expression too large to display}\) | \(5464\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{d + e x + f x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}}{f\,x^2+e\,x+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________