Optimal. Leaf size=600 \[ \frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (a e^2 g^2+c \left (-3 d^2 g^2+6 d e f g-2 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {a \sqrt {c} x}{(-a)^{3/2}}+1}}{\sqrt {2}}\right )|\frac {2 a g}{a g-\sqrt {-a} \sqrt {c} f}\right )}{3 c^{3/2} e^3 \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 (e f-d g)^2 \sqrt {\frac {g \left (\sqrt {-a}-\sqrt {c} x\right )}{\sqrt {-a} g+\sqrt {c} f}} \sqrt {-\frac {g \left (\sqrt {-a}+\sqrt {c} x\right )}{\sqrt {c} f-\sqrt {-a} g}} \Pi \left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g};\sin ^{-1}\left (\sqrt {\frac {c}{c f+\sqrt {-a} \sqrt {c} g}} \sqrt {f+g x}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{e^3 \sqrt {a+c x^2} \sqrt {\frac {c}{\sqrt {-a} \sqrt {c} g+c f}}}-\frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} (7 e f-3 d g) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {a \sqrt {c} x}{(-a)^{3/2}}+1}}{\sqrt {2}}\right )|\frac {2 a g}{a g-\sqrt {-a} \sqrt {c} f}\right )}{3 \sqrt {c} e^2 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}+\frac {2 g^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c e} \]
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Rubi [A] time = 0.95, antiderivative size = 808, normalized size of antiderivative = 1.35, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {958, 719, 419, 933, 168, 538, 537, 424, 743, 844} \[ -\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right ) (e f-d g)^3}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) (e f-d g)^2}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) (e f-d g)}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {8 \sqrt {-a} f g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}+\frac {2 \sqrt {-a} g \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 c^{3/2} e \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {f+g x} \sqrt {c x^2+a}}{3 c e} \]
Antiderivative was successfully verified.
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Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 743
Rule 844
Rule 933
Rule 958
Rubi steps
\begin {align*} \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx &=\int \left (\frac {g (e f-d g)^2}{e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {(e f-d g)^3}{e^3 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {g (e f-d g) \sqrt {f+g x}}{e^2 \sqrt {a+c x^2}}+\frac {g (f+g x)^{3/2}}{e \sqrt {a+c x^2}}\right ) \, dx\\ &=\frac {g \int \frac {(f+g x)^{3/2}}{\sqrt {a+c x^2}} \, dx}{e}+\frac {(g (e f-d g)) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{e^2}+\frac {\left (g (e f-d g)^2\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^3}+\frac {(e f-d g)^3 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^3}\\ &=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}+\frac {(2 g) \int \frac {\frac {1}{2} \left (3 c f^2-a g^2\right )+2 c f g x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 c e}+\frac {\left ((e f-d g)^3 \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}} \sqrt {1+\frac {\sqrt {c} x}{\sqrt {-a}}} (d+e x) \sqrt {f+g x}} \, dx}{e^3 \sqrt {a+c x^2}}+\frac {\left (2 a g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} e^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 a g (e f-d g)^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {(4 f g) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 e}-\frac {\left (g \left (c f^2+a g^2\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 c e}-\frac {\left (2 (e f-d g)^3 \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {f+\frac {\sqrt {-a} g}{\sqrt {c}}-\frac {\sqrt {-a} g x^2}{\sqrt {c}}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^3 \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 (e f-d g)^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {1-\frac {\sqrt {-a} g x^2}{\sqrt {c} \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (8 a f g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} \sqrt {c} e \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (2 a g \left (c f^2+a g^2\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} c^{3/2} e \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {8 \sqrt {-a} f g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} g \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 c^{3/2} e \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 (e f-d g)^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 9.58, size = 1440, normalized size = 2.40 \[ \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} g^2}{3 c e}+\frac {2 (f+g x)^{3/2} \left (\frac {7 c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f^3}{(f+g x)^2}+\frac {3 i c e^2 \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right ) f^2}{\sqrt {f+g x}}-\frac {14 c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f^2}{f+g x}-\frac {3 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f^2}{(f+g x)^2}-\frac {6 i c d e g \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right ) f}{\sqrt {f+g x}}+7 c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f+\frac {6 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f}{f+g x}+\frac {7 a e^2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f}{(f+g x)^2}+\frac {\sqrt {c} e \left (\sqrt {a} g-i \sqrt {c} f\right ) (7 e f-3 d g) \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {i e \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (i \sqrt {a} e g+\sqrt {c} (6 e f-3 d g)\right ) \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {3 i c d^2 g^2 \sqrt {-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \sqrt {-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}+1} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}-3 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-\frac {3 a d e g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}\right )}{3 c e^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \sqrt {\frac {c (f+g x)^2 \left (\frac {f}{f+g x}-1\right )^2}{g^2}+a}} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 3164, normalized size = 5.27 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^{5/2}}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {\left (f + g x\right )^{\frac {5}{2}}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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