Optimal. Leaf size=175 \[ -\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{e (d+e x)^{5/2}}+\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 \sqrt {d+e x}}-\frac {3 c d \sqrt {c d^2-a e^2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{e^{5/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {662, 664, 660, 205} \begin {gather*} -\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{e (d+e x)^{5/2}}+\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 \sqrt {d+e x}}-\frac {3 c d \sqrt {c d^2-a e^2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 660
Rule 662
Rule 664
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e (d+e x)^{5/2}}+\frac {(3 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{3/2}} \, dx}{2 e}\\ &=\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2 \sqrt {d+e x}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e (d+e x)^{5/2}}-\frac {\left (3 c d \left (c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 e^2}\\ &=\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2 \sqrt {d+e x}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e (d+e x)^{5/2}}-\frac {\left (3 c d \left (c d^2-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{e}\\ &=\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2 \sqrt {d+e x}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e (d+e x)^{5/2}}-\frac {3 c d \sqrt {c d^2-a e^2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{e^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 79, normalized size = 0.45 \begin {gather*} \frac {2 c d ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 16.95, size = 169, normalized size = 0.97 \begin {gather*} \frac {((d+e x) (a e+c d x))^{3/2} \left (-\frac {c d \sqrt {a e+c d x} \left (2 e (a e+c d x)-3 a e^2+3 c d^2\right )}{e^2 \left (-e (a e+c d x)+a e^2-c d^2\right )}-\frac {3 c d \sqrt {c d^2-a e^2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{e^{5/2}}\right )}{(d+e x)^{3/2} (a e+c d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 411, normalized size = 2.35 \begin {gather*} \left [\frac {3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{e}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} e \sqrt {-\frac {c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + 3 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{e}} \arctan \left (\frac {\sqrt {e x + d} \sqrt {\frac {c d^{2} - a e^{2}}{e}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + 3 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}}{e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 314, normalized size = 1.79 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (-3 a c d \,e^{3} x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+3 c^{2} d^{3} e x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )-3 a c \,d^{2} e^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+3 c^{2} d^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+2 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c d e x -\sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,e^{2}+3 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c \,d^{2}\right )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, e^{2}} \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 {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {7}{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 = -0.01 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,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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