Optimal. Leaf size=258 \[ -\frac {2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^{7/2}}+\frac {8 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^{5/2}}-\frac {4 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^{3/2}}+\frac {8 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt {d+e x}}+\frac {2 b^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)} \]
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Rubi [A]
time = 0.07, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {784, 21, 45}
\begin {gather*} -\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^{3/2}}+\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^5 (a+b x) (d+e x)^{5/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^5 (a+b x) (d+e x)^{7/2}}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^5 (a+b x)}+\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 784
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^{9/2}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^{9/2}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{9/2}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{7/2}}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^{5/2}}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^{3/2}}+\frac {b^4}{e^4 \sqrt {d+e x}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^{7/2}}+\frac {8 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^{5/2}}-\frac {4 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^{3/2}}+\frac {8 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt {d+e x}}+\frac {2 b^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 173, normalized size = 0.67 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (5 a^4 e^4+4 a^3 b e^3 (2 d+7 e x)+2 a^2 b^2 e^2 \left (8 d^2+28 d e x+35 e^2 x^2\right )+4 a b^3 e \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )-b^4 \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )\right )}{35 e^5 (a+b x) (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 202, normalized size = 0.78
method | result | size |
risch | \(\frac {2 b^{4} \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{e^{5} \left (b x +a \right )}-\frac {2 \left (140 b^{3} e^{3} x^{3}+70 a \,b^{2} e^{3} x^{2}+350 b^{3} d \,e^{2} x^{2}+28 a^{2} b \,e^{3} x +84 a \,b^{2} d \,e^{2} x +308 b^{3} d^{2} e x +5 a^{3} e^{3}+13 a^{2} b d \,e^{2}+29 a \,b^{2} d^{2} e +93 b^{3} d^{3}\right ) \left (a e -b d \right ) \sqrt {\left (b x +a \right )^{2}}}{35 e^{5} \sqrt {e x +d}\, \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right ) \left (b x +a \right )}\) | \(201\) |
gosper | \(-\frac {2 \left (-35 b^{4} e^{4} x^{4}+140 a \,b^{3} e^{4} x^{3}-280 b^{4} d \,e^{3} x^{3}+70 a^{2} b^{2} e^{4} x^{2}+280 a \,b^{3} d \,e^{3} x^{2}-560 b^{4} d^{2} e^{2} x^{2}+28 a^{3} b \,e^{4} x +56 a^{2} b^{2} d \,e^{3} x +224 a \,b^{3} d^{2} e^{2} x -448 b^{4} d^{3} e x +5 a^{4} e^{4}+8 a^{3} b d \,e^{3}+16 a^{2} b^{2} d^{2} e^{2}+64 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{\frac {7}{2}} e^{5} \left (b x +a \right )^{3}}\) | \(202\) |
default | \(-\frac {2 \left (-35 b^{4} e^{4} x^{4}+140 a \,b^{3} e^{4} x^{3}-280 b^{4} d \,e^{3} x^{3}+70 a^{2} b^{2} e^{4} x^{2}+280 a \,b^{3} d \,e^{3} x^{2}-560 b^{4} d^{2} e^{2} x^{2}+28 a^{3} b \,e^{4} x +56 a^{2} b^{2} d \,e^{3} x +224 a \,b^{3} d^{2} e^{2} x -448 b^{4} d^{3} e x +5 a^{4} e^{4}+8 a^{3} b d \,e^{3}+16 a^{2} b^{2} d^{2} e^{2}+64 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{\frac {7}{2}} e^{5} \left (b x +a \right )^{3}}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 328, normalized size = 1.27 \begin {gather*} -\frac {2 \, {\left (35 \, b^{3} x^{3} e^{3} + 16 \, b^{3} d^{3} + 8 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 35 \, {\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 7 \, {\left (8 \, b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} a}{35 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + d^{3} e^{4}\right )} \sqrt {x e + d}} + \frac {2 \, {\left (35 \, b^{3} x^{4} e^{4} + 128 \, b^{3} d^{4} - 48 \, a b^{2} d^{3} e - 8 \, a^{2} b d^{2} e^{2} - 2 \, a^{3} d e^{3} + 35 \, {\left (8 \, b^{3} d e^{3} - 3 \, a b^{2} e^{4}\right )} x^{3} + 35 \, {\left (16 \, b^{3} d^{2} e^{2} - 6 \, a b^{2} d e^{3} - a^{2} b e^{4}\right )} x^{2} + 7 \, {\left (64 \, b^{3} d^{3} e - 24 \, a b^{2} d^{2} e^{2} - 4 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} x\right )} b}{35 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\right )} \sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 205, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (128 \, b^{4} d^{4} + {\left (35 \, b^{4} x^{4} - 140 \, a b^{3} x^{3} - 70 \, a^{2} b^{2} x^{2} - 28 \, a^{3} b x - 5 \, a^{4}\right )} e^{4} + 8 \, {\left (35 \, b^{4} d x^{3} - 35 \, a b^{3} d x^{2} - 7 \, a^{2} b^{2} d x - a^{3} b d\right )} e^{3} + 16 \, {\left (35 \, b^{4} d^{2} x^{2} - 14 \, a b^{3} d^{2} x - a^{2} b^{2} d^{2}\right )} e^{2} + 64 \, {\left (7 \, b^{4} d^{3} x - a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{35 \, {\left (x^{4} e^{9} + 4 \, d x^{3} e^{8} + 6 \, d^{2} x^{2} e^{7} + 4 \, d^{3} x e^{6} + d^{4} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.66, size = 310, normalized size = 1.20 \begin {gather*} 2 \, \sqrt {x e + d} b^{4} e^{\left (-5\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {2 \, {\left (140 \, {\left (x e + d\right )}^{3} b^{4} d \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (x e + d\right )}^{2} b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) + 28 \, {\left (x e + d\right )} b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 5 \, b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 140 \, {\left (x e + d\right )}^{3} a b^{3} e \mathrm {sgn}\left (b x + a\right ) + 140 \, {\left (x e + d\right )}^{2} a b^{3} d e \mathrm {sgn}\left (b x + a\right ) - 84 \, {\left (x e + d\right )} a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (x e + d\right )}^{2} a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, {\left (x e + d\right )} a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 28 \, {\left (x e + d\right )} a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) - 5 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{35 \, {\left (x e + d\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.00, size = 309, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {10\,a^4\,e^4+16\,a^3\,b\,d\,e^3+32\,a^2\,b^2\,d^2\,e^2+128\,a\,b^3\,d^3\,e-256\,b^4\,d^4}{35\,b\,e^8}-\frac {2\,b^3\,x^4}{e^4}+\frac {x\,\left (56\,a^3\,b\,e^4+112\,a^2\,b^2\,d\,e^3+448\,a\,b^3\,d^2\,e^2-896\,b^4\,d^3\,e\right )}{35\,b\,e^8}+\frac {8\,b^2\,x^3\,\left (a\,e-2\,b\,d\right )}{e^5}+\frac {4\,b\,x^2\,\left (a^2\,e^2+4\,a\,b\,d\,e-8\,b^2\,d^2\right )}{e^6}\right )}{x^4\,\sqrt {d+e\,x}+\frac {a\,d^3\,\sqrt {d+e\,x}}{b\,e^3}+\frac {x^3\,\left (35\,a\,e^8+105\,b\,d\,e^7\right )\,\sqrt {d+e\,x}}{35\,b\,e^8}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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