Optimal. Leaf size=411 \[ \frac {\left (\frac {a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt {c}}-\sqrt {-a} \left (c d^2 f-a e (2 d g+e f)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{a c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}+\frac {\left (\sqrt {-a} \left (c d^2 f-a e (2 d g+e f)\right )+\frac {a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt {c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{a c \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f}}+\frac {e \sqrt {d+e x} \sqrt {f+g x}}{c}+\frac {\sqrt {e} (3 d g+e f) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c \sqrt {g}} \]
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Rubi [A] time = 2.51, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {904, 80, 63, 217, 206, 6725, 93, 208} \begin {gather*} \frac {\left (\frac {a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt {c}}-\sqrt {-a} \left (c d^2 f-a e (2 d g+e f)\right )\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 )}{a c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}+\frac {\left (\sqrt {-a} \left (c d^2 f-a e (2 d g+e f)\right )+\frac {a \left (a e^2 g-c d (d g+2 e f)\right )}{\sqrt {c}}\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 c \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f}}+\frac {e \sqrt {d+e x} \sqrt {f+g x}}{c}+\frac {\sqrt {e} (3 d g+e f) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c \sqrt {g}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 93
Rule 206
Rule 208
Rule 217
Rule 904
Rule 6725
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} \sqrt {f+g x}}{a+c x^2} \, dx &=\frac {\int \frac {c d^2 f-a e (e f+2 d g)-\left (a e^2 g-c d (2 e f+d g)\right ) x}{\sqrt {d+e x} \sqrt {f+g x} \left (a+c x^2\right )} \, dx}{c}+\frac {e \int \frac {e f+2 d g+e g x}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{c}\\ &=\frac {e \sqrt {d+e x} \sqrt {f+g x}}{c}+\frac {\int \left (\frac {-\frac {a \left (-a e^2 g+c d (2 e f+d g)\right )}{\sqrt {c}}+\sqrt {-a} \left (c d^2 f-a e (e f+2 d g)\right )}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {\frac {a \left (-a e^2 g+c d (2 e f+d g)\right )}{\sqrt {c}}+\sqrt {-a} \left (c d^2 f-a e (e f+2 d g)\right )}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}\right ) \, dx}{c}+\frac {(e (e f+3 d g)) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 c}\\ &=\frac {e \sqrt {d+e x} \sqrt {f+g x}}{c}+\frac {(e f+3 d g) \operatorname {Subst}\left (\int \frac {1}{\sqrt {f-\frac {d g}{e}+\frac {g x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{c}+\frac {\left (\frac {a \left (a e^2 g-c d (2 e f+d g)\right )}{\sqrt {c}}+\sqrt {-a} \left (c d^2 f-a e (e f+2 d g)\right )\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 a c}+\frac {\left (\frac {a \left (-a e^2 g+c d (2 e f+d g)\right )}{\sqrt {c}}+\sqrt {-a} \left (c d^2 f-a e (e f+2 d g)\right )\right ) \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 a c}\\ &=\frac {e \sqrt {d+e x} \sqrt {f+g x}}{c}+\frac {(e f+3 d g) \operatorname {Subst}\left (\int \frac {1}{1-\frac {g x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{c}+\frac {\left (\frac {a \left (a e^2 g-c d (2 e f+d g)\right )}{\sqrt {c}}+\sqrt {-a} \left (c d^2 f-a e (e f+2 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 )}{a c}+\frac {\left (\frac {a \left (-a e^2 g+c d (2 e f+d g)\right )}{\sqrt {c}}+\sqrt {-a} \left (c d^2 f-a e (e f+2 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 )}{a c}\\ &=\frac {e \sqrt {d+e x} \sqrt {f+g x}}{c}+\frac {\sqrt {e} (e f+3 d g) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c \sqrt {g}}+\frac {\left (\frac {a \left (a e^2 g-c d (2 e f+d g)\right )}{\sqrt {c}}-\sqrt {-a} \left (c d^2 f-a e (e f+2 d g)\right )\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 )}{a c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}+\frac {\left (\frac {a \left (a e^2 g-c d (2 e f+d g)\right )}{\sqrt {c}}+\sqrt {-a} \left (c d^2 f-a e (e f+2 d g)\right )\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 )}{a c \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g}}\\ \end {align*}
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Mathematica [A] time = 2.44, size = 410, normalized size = 1.00 \begin {gather*} \frac {-\frac {\left (\sqrt {-a} c d^2+2 a \sqrt {c} d e+(-a)^{3/2} e^2\right ) \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 )}{a \sqrt {\sqrt {-a} e-\sqrt {c} d}}+\frac {\left (\sqrt {-a} c d^2-2 a \sqrt {c} d e+(-a)^{3/2} e^2\right ) \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 )}{a \sqrt {\sqrt {-a} e+\sqrt {c} d}}+\sqrt {c} e \sqrt {d+e x} \sqrt {f+g x}+\frac {\sqrt {c} \sqrt {e f-d g} (3 d g+e f) \sqrt {\frac {e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )}{\sqrt {g} \sqrt {f+g x}}}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 1.44, size = 580, normalized size = 1.41 \begin {gather*} \frac {\left (-i c d f \sqrt {a e^2+c d^2}+\sqrt {a} \sqrt {c} e f \sqrt {a e^2+c d^2}+\sqrt {a} \sqrt {c} d g \sqrt {a e^2+c d^2}+i a e g \sqrt {a e^2+c d^2}\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} c^{3/2} \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )}}+\frac {\left (i c d f \sqrt {a e^2+c d^2}+\sqrt {a} \sqrt {c} e f \sqrt {a e^2+c d^2}+\sqrt {a} \sqrt {c} d g \sqrt {a e^2+c d^2}-i a e g \sqrt {a e^2+c d^2}\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} c^{3/2} \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )}}+\frac {\left (3 d \sqrt {e} g+e^{3/2} f\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{c \sqrt {g}}-\frac {e \sqrt {f+g x} (e f-d g)}{c \sqrt {d+e x} \left (g-\frac {e (f+g x)}{d+e x}\right )} \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 = 0.11, size = 2497, normalized size = 6.08
result 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 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}}{c x^{2} + a}\,{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 (d+e\,x\right )}^{3/2}}{c\,x^2+a} \,d x \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 {\left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}{a + c x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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