Optimal. Leaf size=326 \[ \frac {\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5}+\frac {\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}-\frac {\left (3 a^2 h^4 (f g-e h)+8 c^2 g^3 \left (f g^2-h (e g-d h)\right )+12 a c g h^2 \left (f g^2-h (e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} h^6}-\frac {\left (c g^2+a h^2\right )^{3/2} \left (f g^2-e g h+d h^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{h^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.47, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1668, 829, 858,
223, 212, 739} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right )}{8 \sqrt {c} h^6}-\frac {\left (a h^2+c g^2\right )^{3/2} \left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{h^6}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{8 h^5}+\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^3}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 739
Rule 829
Rule 858
Rule 1668
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx &=\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac {\int \frac {\left (5 c d h^2-5 c h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{g+h x} \, dx}{5 c h^2}\\ &=\frac {\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac {\int \frac {\left (5 a c^2 h^2 \left (f g^2-h (e g-4 d h)\right )-5 c^2 h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {a+c x^2}}{g+h x} \, dx}{20 c^2 h^4}\\ &=\frac {\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5}+\frac {\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac {\int \frac {5 a c^3 h^2 \left (a h^2 \left (5 f g^2-h (5 e g-8 d h)\right )+4 c \left (f g^4-g^2 h (e g-d h)\right )\right )-5 c^3 h \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) x}{(g+h x) \sqrt {a+c x^2}} \, dx}{40 c^3 h^6}\\ &=\frac {\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5}+\frac {\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac {\left (\left (c g^2+a h^2\right )^2 \left (f g^2-e g h+d h^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{h^6}-\frac {\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 h^6}\\ &=\frac {\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5}+\frac {\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}-\frac {\left (\left (c g^2+a h^2\right )^2 \left (f g^2-e g h+d h^2\right )\right ) \text {Subst}\left (\int \frac {1}{c g^2+a h^2-x^2} \, dx,x,\frac {a h-c g x}{\sqrt {a+c x^2}}\right )}{h^6}-\frac {\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 h^6}\\ &=\frac {\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5}+\frac {\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac {f \left (a+c x^2\right )^{5/2}}{5 c h}-\frac {\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} h^6}-\frac {\left (c g^2+a h^2\right )^{3/2} \left (f g^2-e g h+d h^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{h^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.39, size = 360, normalized size = 1.10 \begin {gather*} \frac {\frac {h \sqrt {a+c x^2} \left (24 a^2 f h^4+a c h^2 \left (5 h (-32 e g+32 d h+15 e h x)+f \left (160 g^2-75 g h x+48 h^2 x^2\right )\right )+2 c^2 \left (f \left (60 g^4-30 g^3 h x+20 g^2 h^2 x^2-15 g h^3 x^3+12 h^4 x^4\right )+5 h \left (2 d h \left (6 g^2-3 g h x+2 h^2 x^2\right )+e \left (-12 g^3+6 g^2 h x-4 g h^2 x^2+3 h^3 x^3\right )\right )\right )\right )}{c}-240 \left (-c g^2-a h^2\right )^{3/2} \left (f g^2+h (-e g+d h)\right ) \tan ^{-1}\left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )+\frac {15 \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2+h (-e g+d h)\right )+8 c^2 \left (f g^5+g^3 h (-e g+d h)\right )\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{120 h^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(642\) vs.
\(2(299)=598\).
time = 0.11, size = 643, normalized size = 1.97
method | result | size |
default | \(\frac {\frac {f h \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}+e h \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )-g f \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{h^{2}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (\frac {\left (\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}\right )^{\frac {3}{2}}}{3}-\frac {c g \left (\frac {\left (2 c \left (x +\frac {g}{h}\right )-\frac {2 c g}{h}\right ) \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (a \,h^{2}+c \,g^{2}\right )}{h^{2}}-\frac {4 c^{2} g^{2}}{h^{2}}\right ) \ln \left (\frac {-\frac {c g}{h}+c \left (x +\frac {g}{h}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{h}+\frac {\left (a \,h^{2}+c \,g^{2}\right ) \left (\sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}-\frac {\sqrt {c}\, g \ln \left (\frac {-\frac {c g}{h}+c \left (x +\frac {g}{h}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\right )}{h}-\frac {\left (a \,h^{2}+c \,g^{2}\right ) \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{2} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{2}}\right )}{h^{3}}\) | \(643\) |
risch | \(\text {Expression too large to display}\) | \(1642\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 642 vs.
\(2 (305) = 610\).
time = 0.37, size = 642, normalized size = 1.97 \begin {gather*} -\frac {\sqrt {c x^{2} + a} c f g^{3} x}{2 \, h^{4}} - \frac {\sqrt {c x^{2} + a} c d g x}{2 \, h^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} f g x}{4 \, h^{2}} - \frac {3 \, \sqrt {c x^{2} + a} a f g x}{8 \, h^{2}} - \frac {c^{\frac {3}{2}} f g^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{6}} - \frac {c^{\frac {3}{2}} d g^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{4}} - \frac {3 \, a \sqrt {c} f g^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, h^{4}} - \frac {3 \, a \sqrt {c} d g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, h^{2}} - \frac {3 \, a^{2} f g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c} h^{2}} + \frac {{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} f g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{3}} + \frac {{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} d \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h} + \frac {\sqrt {c x^{2} + a} c g^{2} x e}{2 \, h^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} x e}{4 \, h} + \frac {3 \, \sqrt {c x^{2} + a} a x e}{8 \, h} + \frac {c^{\frac {3}{2}} g^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e}{h^{5}} + \frac {3 \, a \sqrt {c} g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e}{2 \, h^{3}} + \frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e}{8 \, \sqrt {c} h} - \frac {{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right ) e}{h^{2}} + \frac {\sqrt {c x^{2} + a} c f g^{4}}{h^{5}} + \frac {\sqrt {c x^{2} + a} c d g^{2}}{h^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} f g^{2}}{3 \, h^{3}} + \frac {\sqrt {c x^{2} + a} a f g^{2}}{h^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d}{3 \, h} + \frac {\sqrt {c x^{2} + a} a d}{h} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} f}{5 \, c h} - \frac {\sqrt {c x^{2} + a} c g^{3} e}{h^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} g e}{3 \, h^{2}} - \frac {\sqrt {c x^{2} + a} a g e}{h^{2}} \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}} \left (d + e x + f x^{2}\right )}{g + h x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.79, size = 551, normalized size = 1.69 \begin {gather*} \frac {1}{120} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, {\left (\frac {4 \, c f x}{h} - \frac {5 \, {\left (c^{4} f g h^{19} - c^{4} h^{20} e\right )}}{c^{3} h^{21}}\right )} x + \frac {4 \, {\left (5 \, c^{4} f g^{2} h^{18} + 5 \, c^{4} d h^{20} + 6 \, a c^{3} f h^{20} - 5 \, c^{4} g h^{19} e\right )}}{c^{3} h^{21}}\right )} x - \frac {15 \, {\left (4 \, c^{4} f g^{3} h^{17} + 4 \, c^{4} d g h^{19} + 5 \, a c^{3} f g h^{19} - 4 \, c^{4} g^{2} h^{18} e - 5 \, a c^{3} h^{20} e\right )}}{c^{3} h^{21}}\right )} x + \frac {8 \, {\left (15 \, c^{4} f g^{4} h^{16} + 15 \, c^{4} d g^{2} h^{18} + 20 \, a c^{3} f g^{2} h^{18} + 20 \, a c^{3} d h^{20} + 3 \, a^{2} c^{2} f h^{20} - 15 \, c^{4} g^{3} h^{17} e - 20 \, a c^{3} g h^{19} e\right )}}{c^{3} h^{21}}\right )} + \frac {2 \, {\left (c^{2} f g^{6} + c^{2} d g^{4} h^{2} + 2 \, a c f g^{4} h^{2} + 2 \, a c d g^{2} h^{4} + a^{2} f g^{2} h^{4} + a^{2} d h^{6} - c^{2} g^{5} h e - 2 \, a c g^{3} h^{3} e - a^{2} g h^{5} e\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} h + \sqrt {c} g}{\sqrt {-c g^{2} - a h^{2}}}\right )}{\sqrt {-c g^{2} - a h^{2}} h^{6}} + \frac {{\left (8 \, c^{\frac {5}{2}} f g^{5} + 8 \, c^{\frac {5}{2}} d g^{3} h^{2} + 12 \, a c^{\frac {3}{2}} f g^{3} h^{2} + 12 \, a c^{\frac {3}{2}} d g h^{4} + 3 \, a^{2} \sqrt {c} f g h^{4} - 8 \, c^{\frac {5}{2}} g^{4} h e - 12 \, a c^{\frac {3}{2}} g^{2} h^{3} e - 3 \, a^{2} \sqrt {c} h^{5} e\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c h^{6}} \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}\,\left (f\,x^2+e\,x+d\right )}{g+h\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________