Optimal. Leaf size=204 \[ \frac {2 \sqrt {f} \sqrt {a+b x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|-\frac {b (d e-c f)}{(b c-a d) f}\right )}{\sqrt {e+f x} (b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}}}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d) (b e-a f)} \]
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Rubi [A] time = 0.14, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {104, 21, 114, 113} \[ \frac {2 \sqrt {f} \sqrt {a+b x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|-\frac {b (d e-c f)}{(b c-a d) f}\right )}{\sqrt {e+f x} (b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}}}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 21
Rule 104
Rule 113
Rule 114
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx &=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \int \frac {-\frac {1}{2} a d f-\frac {1}{2} b d f x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{(b c-a d) (b e-a f)}\\ &=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {(d f) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{(b c-a d) (b e-a f)}\\ &=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {\left (d f \sqrt {a+b x} \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {\sqrt {\frac {a d}{-b c+a d}+\frac {b d x}{-b c+a d}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}} \, dx}{(b c-a d) (b e-a f) \sqrt {\frac {d (a+b x)}{-b c+a d}} \sqrt {e+f x}}\\ &=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {f} \sqrt {-d e+c f} \sqrt {a+b x} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|-\frac {b (d e-c f)}{(b c-a d) f}\right )}{(b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}} \sqrt {e+f x}}\\ \end {align*}
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Mathematica [C] time = 1.35, size = 201, normalized size = 0.99 \[ \frac {2 b \sqrt {c+d x} \sqrt {e+f x} \left (-1-\frac {i \sqrt {\frac {d (a+b x)}{b (c+d x)}} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {d (a+b x)}{b c-a d}}\right )|\frac {b c f-a d f}{b d e-a d f}\right )-\operatorname {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {\frac {d (a+b x)}{b c-a d}}\right ),\frac {b c f-a d f}{b d e-a d f}\right )\right )}{\sqrt {\frac {b (e+f x)}{b e-a f}}}\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{2} d f x^{4} + a^{2} c e + {\left (b^{2} d e + {\left (b^{2} c + 2 \, a b d\right )} f\right )} x^{3} + {\left ({\left (b^{2} c + 2 \, a b d\right )} e + {\left (2 \, a b c + a^{2} d\right )} f\right )} x^{2} + {\left (a^{2} c f + {\left (2 \, a b c + a^{2} d\right )} e\right )} x}, x\right ) \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1011, normalized size = 4.96 \[ \frac {2 \left (-b^{2} d f \,x^{2}-\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, a^{2} d f \EllipticE \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )+\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, a^{2} d f \EllipticF \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )+\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, a b c f \EllipticE \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )-\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, a b c f \EllipticF \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )+\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, a b d e \EllipticE \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )-\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, a b d e \EllipticF \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )-\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, b^{2} c e \EllipticE \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )+\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, b^{2} c e \EllipticF \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )-b^{2} c f x -b^{2} d e x -b^{2} c e \right ) \sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {b x +a}}{\left (a f -b e \right ) \left (a d -b c \right ) \left (b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e \right ) b} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x} \sqrt {e + f x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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