Optimal. Leaf size=107 \[ \frac {4 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac {2 (d+e x)^{5/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac {4 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac {2 (d+e x)^{5/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{5/2}}{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (2 \left (2 c d^2 e-e \left (c d^2+a e^2\right )\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{c d e}\\ &=\frac {4 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 (d+e x)^{5/2}}{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.50 \[ -\frac {2 (d+e x)^{3/2} \left (2 a e^2+c d (d+3 e x)\right )}{3 c^2 d^2 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 131, normalized size = 1.22 \[ -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d e x + c d^{2} + 2 \, a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (c^{4} d^{4} e x^{3} + a^{2} c^{2} d^{3} e^{2} + {\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} x^{2} + {\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} x\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 68, normalized size = 0.64 \[ -\frac {2 \left (c d x +a e \right ) \left (3 c d e x +2 a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 50, normalized size = 0.47 \[ -\frac {2 \, {\left (3 \, c d e x + c d^{2} + 2 \, a e^{2}\right )}}{3 \, {\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt {c d x + a e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 147, normalized size = 1.37 \[ -\frac {\left (\frac {2\,x\,\sqrt {d+e\,x}}{c^3\,d^3}+\frac {\left (\frac {2\,c\,d^2}{3}+\frac {4\,a\,e^2}{3}\right )\,\sqrt {d+e\,x}}{c^4\,d^4\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c^4\,d^5+2\,a\,c^3\,d^3\,e^2\right )}{c^4\,d^4\,e}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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