Optimal. Leaf size=258 \[ \frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}{e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) \sqrt {d+e x}}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^5 (a+b x)} \]
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Rubi [A] time = 0.11, 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 = {770, 21, 43} \begin {gather*} \frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^5 (a+b x)}+\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}{e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
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)^{3/2}} \, 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)^{3/2}} \, 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)^{3/2}} \, 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)^{3/2}}-\frac {4 b (b d-a e)^3}{e^4 \sqrt {d+e x}}+\frac {6 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{3/2}}{e^4}+\frac {b^4 (d+e x)^{5/2}}{e^4}\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}}{e^5 (a+b x) \sqrt {d+e x}}-\frac {8 b (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac {4 b^2 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac {8 b^3 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}+\frac {2 b^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 169, normalized size = 0.66 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-35 a^4 e^4+140 a^3 b e^3 (2 d+e x)+70 a^2 b^2 e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+28 a b^3 e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+b^4 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )\right )}{35 e^5 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 20.00, size = 241, normalized size = 0.93 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-35 a^4 e^4+140 a^3 b e^3 (d+e x)+140 a^3 b d e^3-210 a^2 b^2 d^2 e^2+70 a^2 b^2 e^2 (d+e x)^2-420 a^2 b^2 d e^2 (d+e x)+140 a b^3 d^3 e+420 a b^3 d^2 e (d+e x)+28 a b^3 e (d+e x)^3-140 a b^3 d e (d+e x)^2-35 b^4 d^4-140 b^4 d^3 (d+e x)+70 b^4 d^2 (d+e x)^2+5 b^4 (d+e x)^4-28 b^4 d (d+e x)^3\right )}{35 e^4 \sqrt {d+e x} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 192, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 448 \, a b^{3} d^{3} e - 560 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4} - 4 \, {\left (2 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{3} + 2 \, {\left (8 \, b^{4} d^{2} e^{2} - 28 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (16 \, b^{4} d^{3} e - 56 \, a b^{3} d^{2} e^{2} + 70 \, a^{2} b^{2} d e^{3} - 35 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{6} x + d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 327, normalized size = 1.27 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} e^{30} \mathrm {sgn}\left (b x + a\right ) - 28 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d e^{30} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{2} e^{30} \mathrm {sgn}\left (b x + a\right ) - 140 \, \sqrt {x e + d} b^{4} d^{3} e^{30} \mathrm {sgn}\left (b x + a\right ) + 28 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} e^{31} \mathrm {sgn}\left (b x + a\right ) - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d e^{31} \mathrm {sgn}\left (b x + a\right ) + 420 \, \sqrt {x e + d} a b^{3} d^{2} e^{31} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} e^{32} \mathrm {sgn}\left (b x + a\right ) - 420 \, \sqrt {x e + d} a^{2} b^{2} d e^{32} \mathrm {sgn}\left (b x + a\right ) + 140 \, \sqrt {x e + d} a^{3} b e^{33} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-35\right )} - \frac {2 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 202, normalized size = 0.78 \begin {gather*} -\frac {2 \left (-5 b^{4} e^{4} x^{4}-28 a \,b^{3} e^{4} x^{3}+8 b^{4} d \,e^{3} x^{3}-70 a^{2} b^{2} e^{4} x^{2}+56 a \,b^{3} d \,e^{3} x^{2}-16 b^{4} d^{2} e^{2} x^{2}-140 a^{3} b \,e^{4} x +280 a^{2} b^{2} d \,e^{3} x -224 a \,b^{3} d^{2} e^{2} x +64 b^{4} d^{3} e x +35 a^{4} e^{4}-280 a^{3} b d \,e^{3}+560 a^{2} b^{2} d^{2} e^{2}-448 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{35 \sqrt {e x +d}\, \left (b x +a \right )^{3} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 282, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - {\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} + {\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} a}{5 \, \sqrt {e x + d} e^{4}} + \frac {2 \, {\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 336 \, a b^{2} d^{3} e - 280 \, a^{2} b d^{2} e^{2} + 70 \, a^{3} d e^{3} - {\left (8 \, b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} + {\left (16 \, b^{3} d^{2} e^{2} - 42 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} - {\left (64 \, b^{3} d^{3} e - 168 \, a b^{2} d^{2} e^{2} + 140 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )} b}{35 \, \sqrt {e x + d} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.81, size = 218, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^3\,x^4}{7\,e}-\frac {70\,a^4\,e^4-560\,a^3\,b\,d\,e^3+1120\,a^2\,b^2\,d^2\,e^2-896\,a\,b^3\,d^3\,e+256\,b^4\,d^4}{35\,b\,e^5}+\frac {x\,\left (280\,a^3\,b\,e^4-560\,a^2\,b^2\,d\,e^3+448\,a\,b^3\,d^2\,e^2-128\,b^4\,d^3\,e\right )}{35\,b\,e^5}+\frac {8\,b^2\,x^3\,\left (7\,a\,e-2\,b\,d\right )}{35\,e^2}+\frac {4\,b\,x^2\,\left (35\,a^2\,e^2-28\,a\,b\,d\,e+8\,b^2\,d^2\right )}{35\,e^3}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \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 {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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