Optimal. Leaf size=1125 \[ -\frac {\sqrt {2} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right ) e^2}{\sqrt {c} (e f-d g)^3 \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} g \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right ) e}{(e f-d g)^2 \left (c f^2-b g f+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}+\frac {2 g^2 \sqrt {c x^2+b x+a} e}{(e f-d g)^2 \left (c f^2-b g f+a g^2\right ) \sqrt {f+g x}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g (2 c f-b g) \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {4 g^2 (2 c f-b g) \sqrt {c x^2+b x+a}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 g^2 \sqrt {c x^2+b x+a}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}} \]
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Rubi [A] time = 2.31, antiderivative size = 1125, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {957, 744, 834, 843, 718, 424, 419, 21, 934, 169, 538, 537} \[ -\frac {\sqrt {2} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right ) e^2}{\sqrt {c} (e f-d g)^3 \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} g \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right ) e}{(e f-d g)^2 \left (c f^2-b g f+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}+\frac {2 g^2 \sqrt {c x^2+b x+a} e}{(e f-d g)^2 \left (c f^2-b g f+a g^2\right ) \sqrt {f+g x}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g (2 c f-b g) \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {4 g^2 (2 c f-b g) \sqrt {c x^2+b x+a}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 g^2 \sqrt {c x^2+b x+a}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 169
Rule 419
Rule 424
Rule 537
Rule 538
Rule 718
Rule 744
Rule 834
Rule 843
Rule 934
Rule 957
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx &=\int \left (-\frac {g}{(e f-d g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}-\frac {e g}{(e f-d g)^2 (f+g x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {e^2}{(e f-d g)^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}\right ) \, dx\\ &=\frac {e^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{(e f-d g)^2}-\frac {(e g) \int \frac {1}{(f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{(e f-d g)^2}-\frac {g \int \frac {1}{(f+g x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{e f-d g}\\ &=\frac {2 g^2 \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac {2 e g^2 \sqrt {a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}+\frac {(2 e g) \int \frac {-\frac {c f}{2}-\frac {c g x}{2}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right )}+\frac {(2 g) \int \frac {\frac {1}{2} (-3 c f+2 b g)+\frac {c g x}{2}}{(f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )}+\frac {\left (e^2 \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \int \frac {1}{\sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x} (d+e x) \sqrt {f+g x}} \, dx}{(e f-d g)^2 \sqrt {a+b x+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac {4 g^2 (2 c f-b g) \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {(4 g) \int \frac {\frac {1}{4} c \left (3 c f^2-g (b f+a g)\right )+\frac {1}{2} c g (2 c f-b g) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2}-\frac {(c e g) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right )}-\frac {\left (2 e^2 \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}}} \, dx,x,\sqrt {f+g x}\right )}{(e f-d g)^2 \sqrt {a+b x+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac {4 g^2 (2 c f-b g) \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {(2 c g (2 c f-b g)) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2}+\frac {(c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} e g \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}-\frac {\left (2 e^2 \sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{(e f-d g)^2 \sqrt {a+b x+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac {4 g^2 (2 c f-b g) \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} e g \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} g (2 c f-b g) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\left (2 e^2 \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c (f+g x)}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c x^2}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{(e f-d g)^2 \sqrt {a+b x+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac {4 g^2 (2 c f-b g) \sqrt {a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g (2 c f-b g) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} e g \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} e^2 \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} (e f-d g)^3 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 15.52, size = 14759, normalized size = 13.12 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 27597, normalized size = 24.53 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (f+g\,x\right )}^{5/2}\,\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x\right ) \left (f + g x\right )^{\frac {5}{2}} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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