Optimal. Leaf size=328 \[ -\frac {105 b e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}-\frac {35 e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac {21 e^2}{8 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac {3 e}{4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}+\frac {105 b^{3/2} e^3 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \]
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Rubi [A] time = 0.24, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {770, 21, 51, 63, 208} \[ -\frac {105 b e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}-\frac {35 e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac {21 e^2}{8 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac {105 b^{3/2} e^3 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac {3 e}{4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 21
Rule 51
Rule 63
Rule 208
Rule 770
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 )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {a+b x}{\left (a b+b^2 x\right )^5 (d+e x)^{5/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (3 e \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{8 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2}{8 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2}{8 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2}{8 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 b e^3 (a+b x)}{8 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2}{8 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 b e^3 (a+b x)}{8 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 b e^2 \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2}{8 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 b e^3 (a+b x)}{8 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 b^{3/2} e^3 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 68, normalized size = 0.21 \[ -\frac {2 e^3 (a+b x) \, _2F_1\left (-\frac {3}{2},4;-\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{3 \sqrt {(a+b x)^2} (d+e x)^{3/2} (a e-b d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.52, size = 1840, normalized size = 5.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 691, normalized size = 2.11 \[ -\frac {105 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, {\left (b^{5} d^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{5} e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {315 \, {\left (x e + d\right )}^{4} b^{4} e^{3} - 840 \, {\left (x e + d\right )}^{3} b^{4} d e^{3} + 693 \, {\left (x e + d\right )}^{2} b^{4} d^{2} e^{3} - 144 \, {\left (x e + d\right )} b^{4} d^{3} e^{3} - 16 \, b^{4} d^{4} e^{3} + 840 \, {\left (x e + d\right )}^{3} a b^{3} e^{4} - 1386 \, {\left (x e + d\right )}^{2} a b^{3} d e^{4} + 432 \, {\left (x e + d\right )} a b^{3} d^{2} e^{4} + 64 \, a b^{3} d^{3} e^{4} + 693 \, {\left (x e + d\right )}^{2} a^{2} b^{2} e^{5} - 432 \, {\left (x e + d\right )} a^{2} b^{2} d e^{5} - 96 \, a^{2} b^{2} d^{2} e^{5} + 144 \, {\left (x e + d\right )} a^{3} b e^{6} + 64 \, a^{3} b d e^{6} - 16 \, a^{4} e^{7}}{24 \, {\left (b^{5} d^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{5} e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} b - \sqrt {x e + d} b d + \sqrt {x e + d} a e\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 563, normalized size = 1.72 \[ \frac {\left (315 \left (e x +d \right )^{\frac {3}{2}} b^{5} e^{3} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+315 \sqrt {\left (a e -b d \right ) b}\, b^{4} e^{4} x^{4}+945 \left (e x +d \right )^{\frac {3}{2}} a \,b^{4} e^{3} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+840 \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} e^{4} x^{3}+420 \sqrt {\left (a e -b d \right ) b}\, b^{4} d \,e^{3} x^{3}+945 \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{3} e^{3} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+693 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} e^{4} x^{2}+1134 \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d \,e^{3} x^{2}+63 \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{2} e^{2} x^{2}+315 \left (e x +d \right )^{\frac {3}{2}} a^{3} b^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+144 \sqrt {\left (a e -b d \right ) b}\, a^{3} b \,e^{4} x +954 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} d \,e^{3} x +180 \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d^{2} e^{2} x -18 \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{3} e x -16 \sqrt {\left (a e -b d \right ) b}\, a^{4} e^{4}+208 \sqrt {\left (a e -b d \right ) b}\, a^{3} b d \,e^{3}+165 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} d^{2} e^{2}-50 \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d^{3} e +8 \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{4}\right ) \left (b x +a \right )^{2}}{24 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,x}{{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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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