Optimal. Leaf size=818 \[ -\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right ) e^2}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {c x^2+a} e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} f g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {8 c f g^2 \sqrt {c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 g^2 \sqrt {c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}} \]
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Rubi [A] time = 1.00, antiderivative size = 818, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {958, 745, 835, 844, 719, 424, 419, 21, 933, 168, 538, 537} \[ -\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right ) e^2}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {c x^2+a} e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} f g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {8 c f g^2 \sqrt {c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 g^2 \sqrt {c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 745
Rule 835
Rule 844
Rule 933
Rule 958
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx &=\int \left (-\frac {g}{(e f-d g) (f+g x)^{5/2} \sqrt {a+c x^2}}-\frac {e g}{(e f-d g)^2 (f+g x)^{3/2} \sqrt {a+c x^2}}+\frac {e^2}{(e f-d g)^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx\\ &=\frac {e^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2}-\frac {(e g) \int \frac {1}{(f+g x)^{3/2} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2}-\frac {g \int \frac {1}{(f+g x)^{5/2} \sqrt {a+c x^2}} \, dx}{e f-d g}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {(2 c e g) \int \frac {-\frac {f}{2}-\frac {g x}{2}}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2+a g^2\right )}+\frac {(2 c g) \int \frac {-\frac {3 f}{2}+\frac {g x}{2}}{(f+g x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )}+\frac {\left (e^2 \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 f-d g)^2 \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {(4 c g) \int \frac {\frac {1}{4} \left (3 c f^2-a g^2\right )+c f g x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )^2}-\frac {(c e g) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2+a g^2\right )}-\frac {\left (2 e^2 \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 f-d g)^2 \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {\left (4 c^2 f g\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )^2}+\frac {(c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )}-\frac {\left (2 e^2 \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 f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 a \sqrt {c} e 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} (e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {2 \sqrt {-a} \sqrt {c} e 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 )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 e^2 \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 )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (8 a c^{3/2} 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} (e f-d g) \left (c f^2+a g^2\right )^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 \sqrt {c} g \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} (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} 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 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} e 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 )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \sqrt {c} g \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 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 e^2 \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 )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 8.97, size = 1917, normalized size = 2.34 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 9409, normalized size = 11.50 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (f+g\,x\right )}^{5/2}\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right ) \left (f + g x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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