Optimal. Leaf size=167 \[ -\frac {35 b^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}}+\frac {35 b e^2}{4 \sqrt {d+e x} (b d-a e)^4}+\frac {35 e^2}{12 (d+e x)^{3/2} (b d-a e)^3}+\frac {7 e}{4 (a+b x) (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.08, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {35 b^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}}+\frac {35 b e^2}{4 \sqrt {d+e x} (b d-a e)^4}+\frac {35 e^2}{12 (d+e x)^{3/2} (b d-a e)^3}+\frac {7 e}{4 (a+b x) (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (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)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac {(7 e) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{4 (b d-a e)}\\ &=-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {\left (35 e^2\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{8 (b d-a e)^2}\\ &=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {\left (35 b e^2\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^3}\\ &=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {35 b e^2}{4 (b d-a e)^4 \sqrt {d+e x}}+\frac {\left (35 b^2 e^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 (b d-a e)^4}\\ &=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {35 b e^2}{4 (b d-a e)^4 \sqrt {d+e x}}+\frac {\left (35 b^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 (b d-a e)^4}\\ &=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {35 b e^2}{4 (b d-a e)^4 \sqrt {d+e x}}-\frac {35 b^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.31 \begin {gather*} -\frac {2 e^2 \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{3 (d+e x)^{3/2} (a e-b d)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 223, normalized size = 1.34 \begin {gather*} -\frac {e^2 \left (8 a^3 e^3-56 a^2 b e^2 (d+e x)-24 a^2 b d e^2+24 a b^2 d^2 e-175 a b^2 e (d+e x)^2+112 a b^2 d e (d+e x)-8 b^3 d^3-56 b^3 d^2 (d+e x)-105 b^3 (d+e x)^3+175 b^3 d (d+e x)^2\right )}{12 (d+e x)^{3/2} (b d-a e)^4 (-a e-b (d+e x)+b d)^2}-\frac {35 b^{3/2} e^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 (a e-b d)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1226, normalized size = 7.34
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 295, normalized size = 1.77 \begin {gather*} \frac {35 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (9 \, {\left (x e + d\right )} b e^{2} + b d e^{2} - a e^{3}\right )}}{3 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {11 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{2} - 13 \, \sqrt {x e + d} b^{3} d e^{2} + 13 \, \sqrt {x e + d} a b^{2} e^{3}}{4 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 206, normalized size = 1.23 \begin {gather*} \frac {13 \sqrt {e x +d}\, a \,b^{2} e^{3}}{4 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{2}}-\frac {13 \sqrt {e x +d}\, b^{3} d \,e^{2}}{4 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{2}}+\frac {11 \left (e x +d \right )^{\frac {3}{2}} b^{3} e^{2}}{4 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{2}}+\frac {35 b^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a e -b d \right )^{4} \sqrt {\left (a e -b d \right ) b}}+\frac {6 b \,e^{2}}{\left (a e -b d \right )^{4} \sqrt {e x +d}}-\frac {2 e^{2}}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{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 = 2.54, size = 243, normalized size = 1.46 \begin {gather*} \frac {\frac {175\,b^2\,e^2\,{\left (d+e\,x\right )}^2}{12\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^2}{3\,\left (a\,e-b\,d\right )}+\frac {35\,b^3\,e^2\,{\left (d+e\,x\right )}^3}{4\,{\left (a\,e-b\,d\right )}^4}+\frac {14\,b\,e^2\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{b^2\,{\left (d+e\,x\right )}^{7/2}-\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}+\frac {35\,b^{3/2}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^{9/2}}\right )}{4\,{\left (a\,e-b\,d\right )}^{9/2}} \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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