Optimal. Leaf size=231 \[ -\frac {2 (b+2 c x) \sqrt {d+e x}}{b^2 \sqrt {b x+c x^2}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}} \]
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Rubi [A] time = 0.15, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {736, 843, 715, 112, 110, 117, 116} \[ -\frac {2 (b+2 c x) \sqrt {d+e x}}{b^2 \sqrt {b x+c x^2}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 736
Rule 843
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{b^2 \sqrt {b x+c x^2}}+\frac {2 \int \frac {\frac {b e}{2}+c e x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{b^2}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{b^2 \sqrt {b x+c x^2}}+\frac {(2 c) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{b^2}-\frac {(2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{b^2}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{b^2 \sqrt {b x+c x^2}}+\frac {\left (2 c \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{b^2 \sqrt {b x+c x^2}}-\frac {\left ((2 c d-b e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{b^2 \sqrt {b x+c x^2}}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{b^2 \sqrt {b x+c x^2}}+\frac {\left (2 c \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{b^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left ((2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{b^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{b^2 \sqrt {b x+c x^2}}+\frac {4 \sqrt {c} \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.39, size = 186, normalized size = 0.81 \[ \frac {-2 i e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+4 i e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 \sqrt {\frac {b}{c}} (d+e x)}{b \sqrt {\frac {b}{c}} \sqrt {x (b+c x)} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 352, normalized size = 1.52 \[ \frac {2 \left (-2 c^{2} e \,x^{2}-2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b c d \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b c d \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-b c e x -2 c^{2} d x -b c d \right ) \sqrt {\left (c x +b \right ) x}}{\left (c x +b \right ) \sqrt {e x +d}\, b^{2} c x} \]
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 \[ \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {d + e x}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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