Optimal. Leaf size=133 \[ \frac {16 \sqrt {a+b x} \left (15 a^2 e^2-35 a b d e+23 b^2 d^2\right )}{15 \sqrt {d+e x} (b d-a e)^3}+\frac {6 d^2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}+\frac {8 d \sqrt {a+b x} (8 b d-5 a e)}{15 (d+e x)^{3/2} (b d-a e)^2} \]
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Rubi [A] time = 0.13, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {949, 78, 37} \[ \frac {16 \sqrt {a+b x} \left (15 a^2 e^2-35 a b d e+23 b^2 d^2\right )}{15 \sqrt {d+e x} (b d-a e)^3}+\frac {6 d^2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}+\frac {8 d \sqrt {a+b x} (8 b d-5 a e)}{15 (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 949
Rubi steps
\begin {align*} \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx &=\frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {2 \int \frac {6 d (6 b d-5 a e)+20 e (b d-a e) x}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{5 (b d-a e)}\\ &=\frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {8 d (8 b d-5 a e) \sqrt {a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac {\left (8 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{15 (b d-a e)^2}\\ &=\frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {8 d (8 b d-5 a e) \sqrt {a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac {16 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right ) \sqrt {a+b x}}{15 (b d-a e)^3 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 110, normalized size = 0.83 \[ \frac {2 \sqrt {a+b x} \left (a^2 e^2 \left (149 d^2+260 d e x+120 e^2 x^2\right )-2 a b d e \left (175 d^2+306 d e x+140 e^2 x^2\right )+b^2 d^2 \left (225 d^2+400 d e x+184 e^2 x^2\right )\right )}{15 (d+e x)^{5/2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.27, size = 293, normalized size = 2.20 \[ \frac {2 \, {\left (225 \, b^{2} d^{4} - 350 \, a b d^{3} e + 149 \, a^{2} d^{2} e^{2} + 8 \, {\left (23 \, b^{2} d^{2} e^{2} - 35 \, a b d e^{3} + 15 \, a^{2} e^{4}\right )} x^{2} + 4 \, {\left (100 \, b^{2} d^{3} e - 153 \, a b d^{2} e^{2} + 65 \, a^{2} d e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{15 \, {\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} + {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \]
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 [A] time = 0.01, size = 150, normalized size = 1.13 \[ -\frac {2 \sqrt {b x +a}\, \left (120 a^{2} e^{4} x^{2}-280 a b d \,e^{3} x^{2}+184 b^{2} d^{2} e^{2} x^{2}+260 a^{2} d \,e^{3} x -612 a b \,d^{2} e^{2} x +400 b^{2} d^{3} e x +149 a^{2} d^{2} e^{2}-350 a b \,d^{3} e +225 b^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.30, size = 268, normalized size = 2.02 \[ -\frac {\sqrt {d+e\,x}\,\left (\frac {x^2\,\left (240\,a^3\,e^4-40\,a^2\,b\,d\,e^3-856\,a\,b^2\,d^2\,e^2+800\,b^3\,d^3\,e\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\left (520\,a^3\,d\,e^3-926\,a^2\,b\,d^2\,e^2+100\,a\,b^2\,d^3\,e+450\,b^3\,d^4\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {2\,a\,d^2\,\left (149\,a^2\,e^2-350\,a\,b\,d\,e+225\,b^2\,d^2\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {16\,b\,x^3\,\left (15\,a^2\,e^2-35\,a\,b\,d\,e+23\,b^2\,d^2\right )}{15\,e\,{\left (a\,e-b\,d\right )}^3}\right )}{x^3\,\sqrt {a+b\,x}+\frac {d^3\,\sqrt {a+b\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {a+b\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {a+b\,x}}{e^2}} \]
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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