Optimal. Leaf size=613 \[ \frac {e \sqrt {f+g x} \left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right )}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e \sqrt {f+g x} \left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right )}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right ) (e f-d g)}+\frac {4 e g \sqrt {f+g x}}{3 \sqrt {d+e x} \left (a e^2+c d^2\right ) (e f-d g)}-\frac {2 e \sqrt {f+g x}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac {\sqrt {c} \left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \left (a e^2+c d^2\right ) \sqrt {\sqrt {c} f-\sqrt {-a} g}}+\frac {\sqrt {c} \left (a \sqrt {c} (e f-d g)+\sqrt {-a} c d f+\sqrt {-a} a e g\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{a \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \left (a e^2+c d^2\right ) \sqrt {\sqrt {-a} g+\sqrt {c} f}} \]
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Rubi [A] time = 3.16, antiderivative size = 613, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {908, 45, 37, 6725, 96, 93, 208} \begin {gather*} \frac {e \sqrt {f+g x} \left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right )}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right ) (e f-d g)}-\frac {e \sqrt {f+g x} \left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right )}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right ) (e f-d g)}+\frac {4 e g \sqrt {f+g x}}{3 \sqrt {d+e x} \left (a e^2+c d^2\right ) (e f-d g)}-\frac {2 e \sqrt {f+g x}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac {\sqrt {c} \left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \left (a e^2+c d^2\right ) \sqrt {\sqrt {c} f-\sqrt {-a} g}}+\frac {\sqrt {c} \left (a \sqrt {c} (e f-d g)+\sqrt {-a} c d f+\sqrt {-a} a e g\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{a \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \left (a e^2+c d^2\right ) \sqrt {\sqrt {-a} g+\sqrt {c} f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 93
Rule 96
Rule 208
Rule 908
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt {f+g x}}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx &=\frac {\int \frac {c d f+a e g-c (e f-d g) x}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx}{c d^2+a e^2}+\frac {(e (e f-d g)) \int \frac {1}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx}{c d^2+a e^2}\\ &=-\frac {2 e \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {\int \left (\frac {a \sqrt {c} (e f-d g)+\sqrt {-a} (c d f+a e g)}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}}+\frac {-a \sqrt {c} (e f-d g)+\sqrt {-a} (c d f+a e g)}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}}\right ) \, dx}{c d^2+a e^2}-\frac {(2 e g) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{3 \left (c d^2+a e^2\right )}\\ &=-\frac {2 e \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 e g \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}-\frac {\left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}} \, dx}{2 \sqrt {-a} \left (c d^2+a e^2\right )}-\frac {\left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}} \, dx}{2 \sqrt {-a} \left (c d^2+a e^2\right )}\\ &=-\frac {2 e \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 e g \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}+\frac {e \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}-\frac {e \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}-\frac {\left (\sqrt {c} \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right )\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )}-\frac {\left (\sqrt {c} \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right )\right ) \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right )}\\ &=-\frac {2 e \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 e g \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}+\frac {e \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}-\frac {e \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}-\frac {\left (\sqrt {c} \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c} d+\sqrt {-a} e-\left (\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )}-\frac {\left (\sqrt {c} \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {c} d+\sqrt {-a} e-\left (-\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right )}\\ &=-\frac {2 e \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 e g \sqrt {f+g x}}{3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}+\frac {e \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}-\frac {e \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {d+e x}}+\frac {\sqrt {c} \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \left (c d^2+a e^2\right ) \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {c} \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right )^{3/2} \left (c d^2+a e^2\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g}}\\ \end {align*}
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Mathematica [A] time = 2.86, size = 353, normalized size = 0.58 \begin {gather*} -\frac {2 e \sqrt {f+g x} \left (a e^3 (f+g x)+c d \left (-6 d^2 g+7 d e f-5 d e g x+6 e^2 f x\right )\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (e f-d g)}-\frac {\sqrt {c} \sqrt {\sqrt {-a} g-\sqrt {c} f} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g-\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e-\sqrt {c} d}}\right )}{\left (\sqrt {-a} e-\sqrt {c} d\right )^{3/2} \left (\sqrt {-a} \sqrt {c} d+a e\right )}-\frac {\sqrt {c} \sqrt {\sqrt {-a} g+\sqrt {c} f} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \left (\sqrt {-a} \sqrt {c} d-a e\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 1.78, size = 498, normalized size = 0.81 \begin {gather*} \frac {\left (\sqrt {c} d-i \sqrt {a} e\right )^3 \left (\sqrt {a} \sqrt {c} g-i c f\right ) \tan ^{-1}\left (\frac {\sqrt {f+g x} \sqrt {a e^2+c d^2}}{\sqrt {d+e x} \sqrt {-i \sqrt {a} \sqrt {c} d g+i \sqrt {a} \sqrt {c} e f-a e g-c d f}}\right )}{\sqrt {a} \left (a e^2+c d^2\right )^{5/2} \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )}}+\frac {\left (\sqrt {c} d+i \sqrt {a} e\right )^3 \left (\sqrt {a} \sqrt {c} g+i c f\right ) \tan ^{-1}\left (\frac {\sqrt {f+g x} \sqrt {a e^2+c d^2}}{\sqrt {d+e x} \sqrt {i \sqrt {a} \sqrt {c} d g-i \sqrt {a} \sqrt {c} e f-a e g-c d f}}\right )}{\sqrt {a} \left (a e^2+c d^2\right )^{5/2} \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )}}+\frac {2 \left (\frac {a e^4 (f+g x)^{3/2}}{(d+e x)^{3/2}}+\frac {c d^2 e^2 (f+g x)^{3/2}}{(d+e x)^{3/2}}-\frac {6 c d^2 e g \sqrt {f+g x}}{\sqrt {d+e x}}+\frac {6 c d e^2 f \sqrt {f+g x}}{\sqrt {d+e x}}\right )}{3 \left (a e^2+c d^2\right )^2 (d g-e f)} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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.
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giac [F(-1)] 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.
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maple [B] time = 0.09, size = 14861, normalized size = 24.24 \begin {gather*} \text {output too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {g x + f}}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \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 {\sqrt {f+g\,x}}{\left (c\,x^2+a\right )\,{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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.
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