Optimal. Leaf size=225 \[ -\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\left (2 a h^2 (2 f g-e h)+c g \left (f g^2+h (e g-3 d h)\right )\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{2 \left (c g^2+a h^2\right )^{5/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1665, 821, 739,
212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\right )+2 c^2 d g^2\right )}{2 \left (a h^2+c g^2\right )^{5/2}}-\frac {\sqrt {a+c x^2} \left (d h^2-e g h+f g^2\right )}{2 h (g+h x)^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {a+c x^2} \left (2 a h^2 (2 f g-e h)+c g h (e g-3 d h)+c f g^3\right )}{2 h (g+h x) \left (a h^2+c g^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 821
Rule 1665
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2}{(g+h x)^3 \sqrt {a+c x^2}} \, dx &=-\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\int \frac {-2 (c d g-a f g+a e h)-\left (2 a f h+c \left (e g+\frac {f g^2}{h}-d h\right )\right ) x}{(g+h x)^2 \sqrt {a+c x^2}} \, dx}{2 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\left (c f g^3+c g h (e g-3 d h)+2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{2 \left (c g^2+a h^2\right )^2}\\ &=-\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\left (c f g^3+c g h (e g-3 d h)+2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\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 )}{2 \left (c g^2+a h^2\right )^2}\\ &=-\frac {\left (f g^2-e g h+d h^2\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac {\left (c f g^3+c g h (e g-3 d h)+2 a h^2 (2 f g-e h)\right ) \sqrt {a+c x^2}}{2 h \left (c g^2+a h^2\right )^2 (g+h x)}-\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2-h (3 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{2 \left (c g^2+a h^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.20, size = 203, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a+c x^2} \left (c g \left (f g^2 x+e g (2 g+h x)-d h (4 g+3 h x)\right )-a h (-f g (3 g+4 h x)+h (d h+e (g+2 h x)))\right )}{2 \left (c g^2+a h^2\right )^2 (g+h x)^2}-\frac {\left (2 c^2 d g^2+2 a^2 f h^2-a c \left (f g^2+h (-3 e g+d h)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )}{\left (-c g^2-a h^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(809\) vs.
\(2(209)=418\).
time = 0.09, size = 810, normalized size = 3.60
method | result | size |
default | \(-\frac {f \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^{3} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {\left (e h -2 g f \right ) \left (-\frac {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}}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \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 )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{4}}+\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (-\frac {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}}}}{2 \left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )^{2}}+\frac {3 c g h \left (-\frac {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}}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \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 )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{2 \left (a \,h^{2}+c \,g^{2}\right )}+\frac {c \,h^{2} \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 )}{2 \left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{5}}\) | \(810\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 901 vs.
\(2 (214) = 428\).
time = 0.32, size = 901, normalized size = 4.00 \begin {gather*} -\frac {3 \, \sqrt {c x^{2} + a} c f g^{3}}{2 \, {\left (c^{2} g^{4} h^{2} x + 2 \, a c g^{2} h^{4} x + a^{2} h^{6} x + c^{2} g^{5} h + 2 \, a c g^{3} h^{3} + a^{2} g h^{5}\right )}} + \frac {3 \, \sqrt {c x^{2} + a} c g^{2} e}{2 \, {\left (c^{2} g^{4} h x + 2 \, a c g^{2} h^{3} x + a^{2} h^{5} x + c^{2} g^{5} + 2 \, a c g^{3} h^{2} + a^{2} g h^{4}\right )}} - \frac {3 \, \sqrt {c x^{2} + a} c d g}{2 \, {\left (c^{2} g^{4} x + 2 \, a c g^{2} h^{2} x + a^{2} h^{4} x + \frac {c^{2} g^{5}}{h} + 2 \, a c g^{3} h + a^{2} g h^{3}\right )}} - \frac {\sqrt {c x^{2} + a} f g^{2}}{2 \, {\left (c g^{2} h^{3} x^{2} + a h^{5} x^{2} + 2 \, c g^{3} h^{2} x + 2 \, a g h^{4} x + c g^{4} h + a g^{2} h^{3}\right )}} + \frac {2 \, \sqrt {c x^{2} + a} f g}{c g^{2} h^{2} x + a h^{4} x + c g^{3} h + a g h^{3}} + \frac {\sqrt {c x^{2} + a} g e}{2 \, {\left (c g^{2} h^{2} x^{2} + a h^{4} x^{2} + 2 \, c g^{3} h x + 2 \, a g h^{3} x + c g^{4} + a g^{2} h^{2}\right )}} - \frac {\sqrt {c x^{2} + a} d}{2 \, {\left (c g^{2} h x^{2} + a h^{3} x^{2} + 2 \, c g^{3} x + 2 \, a g h^{2} x + \frac {c g^{4}}{h} + a g^{2} h\right )}} - \frac {\sqrt {c x^{2} + a} e}{c g^{2} h x + a h^{3} x + c g^{3} + a g h^{2}} + \frac {3 \, c^{2} f g^{4} \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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{7}} + \frac {3 \, c^{2} d 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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{5}} - \frac {5 \, c 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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{5}} - \frac {c 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 )}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} + \frac {f \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 )}{\sqrt {a + \frac {c g^{2}}{h^{2}}} h^{3}} - \frac {3 \, c^{2} g^{3} \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}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {5}{2}} h^{6}} + \frac {3 \, c 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}{2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 534 vs.
\(2 (214) = 428\).
time = 3.96, size = 1095, normalized size = 4.87 \begin {gather*} \left [\frac {{\left ({\left (2 \, c^{2} d - a c f\right )} g^{4} - {\left (a c d - 2 \, a^{2} f\right )} g^{2} h^{2} + {\left ({\left (2 \, c^{2} d - a c f\right )} g^{2} h^{2} - {\left (a c d - 2 \, a^{2} f\right )} h^{4}\right )} x^{2} + 2 \, {\left ({\left (2 \, c^{2} d - a c f\right )} g^{3} h - {\left (a c d - 2 \, a^{2} f\right )} g h^{3}\right )} x + 3 \, {\left (a c g h^{3} x^{2} + 2 \, a c g^{2} h^{2} x + a c g^{3} h\right )} e\right )} \sqrt {c g^{2} + a h^{2}} \log \left (\frac {2 \, a c g h x - a c g^{2} - 2 \, a^{2} h^{2} - {\left (2 \, c^{2} g^{2} + a c h^{2}\right )} x^{2} - 2 \, \sqrt {c g^{2} + a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{h^{2} x^{2} + 2 \, g h x + g^{2}}\right ) - 2 \, {\left (a^{2} d h^{5} + {\left (4 \, c^{2} d - 3 \, a c f\right )} g^{4} h + {\left (5 \, a c d - 3 \, a^{2} f\right )} g^{2} h^{3} - {\left (c^{2} f g^{5} - {\left (3 \, c^{2} d - 5 \, a c f\right )} g^{3} h^{2} - {\left (3 \, a c d - 4 \, a^{2} f\right )} g h^{4}\right )} x - {\left (2 \, c^{2} g^{5} + a c g^{3} h^{2} - a^{2} g h^{4} + {\left (c^{2} g^{4} h - a c g^{2} h^{3} - 2 \, a^{2} h^{5}\right )} x\right )} e\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} g^{8} + 3 \, a c^{2} g^{6} h^{2} + 3 \, a^{2} c g^{4} h^{4} + a^{3} g^{2} h^{6} + {\left (c^{3} g^{6} h^{2} + 3 \, a c^{2} g^{4} h^{4} + 3 \, a^{2} c g^{2} h^{6} + a^{3} h^{8}\right )} x^{2} + 2 \, {\left (c^{3} g^{7} h + 3 \, a c^{2} g^{5} h^{3} + 3 \, a^{2} c g^{3} h^{5} + a^{3} g h^{7}\right )} x\right )}}, -\frac {{\left ({\left (2 \, c^{2} d - a c f\right )} g^{4} - {\left (a c d - 2 \, a^{2} f\right )} g^{2} h^{2} + {\left ({\left (2 \, c^{2} d - a c f\right )} g^{2} h^{2} - {\left (a c d - 2 \, a^{2} f\right )} h^{4}\right )} x^{2} + 2 \, {\left ({\left (2 \, c^{2} d - a c f\right )} g^{3} h - {\left (a c d - 2 \, a^{2} f\right )} g h^{3}\right )} x + 3 \, {\left (a c g h^{3} x^{2} + 2 \, a c g^{2} h^{2} x + a c g^{3} h\right )} e\right )} \sqrt {-c g^{2} - a h^{2}} \arctan \left (\frac {\sqrt {-c g^{2} - a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{a c g^{2} + a^{2} h^{2} + {\left (c^{2} g^{2} + a c h^{2}\right )} x^{2}}\right ) + {\left (a^{2} d h^{5} + {\left (4 \, c^{2} d - 3 \, a c f\right )} g^{4} h + {\left (5 \, a c d - 3 \, a^{2} f\right )} g^{2} h^{3} - {\left (c^{2} f g^{5} - {\left (3 \, c^{2} d - 5 \, a c f\right )} g^{3} h^{2} - {\left (3 \, a c d - 4 \, a^{2} f\right )} g h^{4}\right )} x - {\left (2 \, c^{2} g^{5} + a c g^{3} h^{2} - a^{2} g h^{4} + {\left (c^{2} g^{4} h - a c g^{2} h^{3} - 2 \, a^{2} h^{5}\right )} x\right )} e\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} g^{8} + 3 \, a c^{2} g^{6} h^{2} + 3 \, a^{2} c g^{4} h^{4} + a^{3} g^{2} h^{6} + {\left (c^{3} g^{6} h^{2} + 3 \, a c^{2} g^{4} h^{4} + 3 \, a^{2} c g^{2} h^{6} + a^{3} h^{8}\right )} x^{2} + 2 \, {\left (c^{3} g^{7} h + 3 \, a c^{2} g^{5} h^{3} + 3 \, a^{2} c g^{3} h^{5} + a^{3} g h^{7}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d + e x + f x^{2}}{\sqrt {a + c x^{2}} \left (g + h x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 848 vs.
\(2 (214) = 428\).
time = 2.83, size = 848, normalized size = 3.77 \begin {gather*} -\frac {{\left (2 \, c^{2} d g^{2} - a c f g^{2} - a c d h^{2} + 2 \, a^{2} f h^{2} + 3 \, a c g h 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 )}{{\left (c^{2} g^{4} + 2 \, a c g^{2} h^{2} + a^{2} h^{4}\right )} \sqrt {-c g^{2} - a h^{2}}} + \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} f g^{4} h - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{2} d g^{2} h^{3} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c f g^{2} h^{3} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c d h^{5} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c g h^{4} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} f g^{5} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} d g^{3} h^{2} + 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} f g^{3} h^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} d g h^{4} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} \sqrt {c} f g h^{4} + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {5}{2}} g^{4} h e - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {3}{2}} g^{2} h^{3} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} \sqrt {c} h^{5} e - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} f g^{4} h + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} d g^{2} h^{3} - 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c f g^{2} h^{3} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c d h^{5} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c^{2} g^{3} h^{2} e + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c g h^{4} e + a^{2} c^{\frac {3}{2}} f g^{3} h^{2} - 3 \, a^{2} c^{\frac {3}{2}} d g h^{4} + 4 \, a^{3} \sqrt {c} f g h^{4} + a^{2} c^{\frac {3}{2}} g^{2} h^{3} e - 2 \, a^{3} \sqrt {c} h^{5} e}{{\left (c^{2} g^{4} h^{2} + 2 \, a c g^{2} h^{4} + a^{2} h^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} h + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} g - a h\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {f\,x^2+e\,x+d}{{\left (g+h\,x\right )}^3\,\sqrt {c\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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