Optimal. Leaf size=233 \[ \frac {32 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^4 d^4 \sqrt {d+e x}}+\frac {16 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^3 d^3}+\frac {12 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d} \]
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Rubi [A] time = 0.18, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \begin {gather*} \frac {32 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^4 d^4 \sqrt {d+e x}}+\frac {16 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^3 d^3}+\frac {12 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac {\left (6 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 d}\\ &=\frac {12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac {\left (24 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3}+\frac {12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 d^3}\\ &=\frac {32 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^4 d^4 \sqrt {d+e x}}+\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3}+\frac {12 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2}+\frac {2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 131, normalized size = 0.56 \begin {gather*} \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (-16 a^3 e^6+8 a^2 c d e^4 (7 d+e x)-2 a c^2 d^2 e^2 \left (35 d^2+14 d e x+3 e^2 x^2\right )+c^3 d^3 \left (35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3\right )\right )}{35 c^4 d^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 194, normalized size = 0.83 \begin {gather*} \frac {2 \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}} \left (-16 a^3 e^6+48 a^2 c d^2 e^4+8 a^2 c d e^4 (d+e x)-48 a c^2 d^4 e^2-16 a c^2 d^3 e^2 (d+e x)-6 a c^2 d^2 e^2 (d+e x)^2+16 c^3 d^6+8 c^3 d^5 (d+e x)+6 c^3 d^4 (d+e x)^2+5 c^3 d^3 (d+e x)^3\right )}{35 c^4 d^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 173, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (5 \, c^{3} d^{3} e^{3} x^{3} + 35 \, c^{3} d^{6} - 70 \, a c^{2} d^{4} e^{2} + 56 \, a^{2} c d^{2} e^{4} - 16 \, a^{3} e^{6} + 3 \, {\left (7 \, c^{3} d^{4} e^{2} - 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (35 \, c^{3} d^{5} e - 28 \, a c^{2} d^{3} e^{3} + 8 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{35 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 168, normalized size = 0.72 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-5 c^{3} d^{3} e^{3} x^{3}+6 a \,c^{2} d^{2} e^{4} x^{2}-21 c^{3} d^{4} e^{2} x^{2}-8 a^{2} c d \,e^{5} x +28 a \,c^{2} d^{3} e^{3} x -35 c^{3} d^{5} e x +16 a^{3} e^{6}-56 a^{2} c \,d^{2} e^{4}+70 a \,c^{2} d^{4} e^{2}-35 c^{3} d^{6}\right ) \sqrt {e x +d}}{35 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 192, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (5 \, c^{4} d^{4} e^{3} x^{4} + 35 \, a c^{3} d^{6} e - 70 \, a^{2} c^{2} d^{4} e^{3} + 56 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + {\left (21 \, c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{3} + {\left (35 \, c^{4} d^{6} e - 7 \, a c^{3} d^{4} e^{3} + 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (35 \, c^{4} d^{7} - 35 \, a c^{3} d^{5} e^{2} + 28 \, a^{2} c^{2} d^{3} e^{4} - 8 \, a^{3} c d e^{6}\right )} x\right )}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 194, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (32\,a^3\,e^6-112\,a^2\,c\,d^2\,e^4+140\,a\,c^2\,d^4\,e^2-70\,c^3\,d^6\right )}{35\,c^4\,d^4\,e}-\frac {2\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^4-28\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{35\,c^3\,d^3}-\frac {2\,e^2\,x^3\,\sqrt {d+e\,x}}{7\,c\,d}+\frac {6\,e\,x^2\,\left (2\,a\,e^2-7\,c\,d^2\right )\,\sqrt {d+e\,x}}{35\,c^2\,d^2}\right )}{x+\frac {d}{e}} \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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