Optimal. Leaf size=119 \[ -\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}+\frac {2 b^2}{\sqrt {d+e x} (b d-a e)^3}+\frac {2 b}{3 (d+e x)^{3/2} (b d-a e)^2}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)} \]
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Rubi [A] time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \begin {gather*} \frac {2 b^2}{\sqrt {d+e x} (b d-a e)^3}-\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}+\frac {2 b}{3 (d+e x)^{3/2} (b d-a e)^2}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx\\ &=\frac {2}{5 (b d-a e) (d+e x)^{5/2}}+\frac {b \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{b d-a e}\\ &=\frac {2}{5 (b d-a e) (d+e x)^{5/2}}+\frac {2 b}{3 (b d-a e)^2 (d+e x)^{3/2}}+\frac {b^2 \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{(b d-a e)^2}\\ &=\frac {2}{5 (b d-a e) (d+e x)^{5/2}}+\frac {2 b}{3 (b d-a e)^2 (d+e x)^{3/2}}+\frac {2 b^2}{(b d-a e)^3 \sqrt {d+e x}}+\frac {b^3 \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{(b d-a e)^3}\\ &=\frac {2}{5 (b d-a e) (d+e x)^{5/2}}+\frac {2 b}{3 (b d-a e)^2 (d+e x)^{3/2}}+\frac {2 b^2}{(b d-a e)^3 \sqrt {d+e x}}+\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e (b d-a e)^3}\\ &=\frac {2}{5 (b d-a e) (d+e x)^{5/2}}+\frac {2 b}{3 (b d-a e)^2 (d+e x)^{3/2}}+\frac {2 b^2}{(b d-a e)^3 \sqrt {d+e x}}-\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 48, normalized size = 0.40 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{5 (d+e x)^{5/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 137, normalized size = 1.15 \begin {gather*} \frac {2 \left (3 a^2 e^2-5 a b e (d+e x)-6 a b d e+3 b^2 d^2+15 b^2 (d+e x)^2+5 b^2 d (d+e x)\right )}{15 (d+e x)^{5/2} (b d-a e)^3}+\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{(a e-b d)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 706, normalized size = 5.93 \begin {gather*} \left [-\frac {15 \, {\left (b^{2} e^{3} x^{3} + 3 \, b^{2} d e^{2} x^{2} + 3 \, b^{2} d^{2} e x + b^{2} d^{3}\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) - 2 \, {\left (15 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 11 \, a b d e + 3 \, a^{2} e^{2} + 5 \, {\left (7 \, b^{2} d e - a b e^{2}\right )} x\right )} \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 )}}, -\frac {2 \, {\left (15 \, {\left (b^{2} e^{3} x^{3} + 3 \, b^{2} d e^{2} x^{2} + 3 \, b^{2} d^{2} e x + b^{2} d^{3}\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (15 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 11 \, a b d e + 3 \, a^{2} e^{2} + 5 \, {\left (7 \, b^{2} d e - a b e^{2}\right )} x\right )} \sqrt {e x + d}\right )}}{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 )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 189, normalized size = 1.59 \begin {gather*} \frac {2 \, b^{3} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} b^{2} + 5 \, {\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} - 5 \, {\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}\right )}}{15 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 112, normalized size = 0.94 \begin {gather*} -\frac {2 b^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3} \sqrt {\left (a e -b d \right ) b}}-\frac {2 b^{2}}{\left (a e -b d \right )^{3} \sqrt {e x +d}}+\frac {2 b}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2}{5 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}}} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 137, normalized size = 1.15 \begin {gather*} -\frac {\frac {2}{5\,\left (a\,e-b\,d\right )}+\frac {2\,b^2\,{\left (d+e\,x\right )}^2}{{\left (a\,e-b\,d\right )}^3}-\frac {2\,b\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{5/2}}-\frac {2\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^{7/2}}\right )}{{\left (a\,e-b\,d\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 140.18, size = 109, normalized size = 0.92 \begin {gather*} - \frac {2 b^{2}}{\sqrt {d + e x} \left (a e - b d\right )^{3}} - \frac {2 b^{2} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{\sqrt {\frac {a e - b d}{b}} \left (a e - b d\right )^{3}} + \frac {2 b}{3 \left (d + e x\right )^{\frac {3}{2}} \left (a e - b d\right )^{2}} - \frac {2}{5 \left (d + e x\right )^{\frac {5}{2}} \left (a e - b d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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