Optimal. Leaf size=254 \[ \frac {35 e^2 (b d-a e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 52, 65,
214} \begin {gather*} -\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) \sqrt {d+e x} (b d-a e)}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 660
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{8 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{8 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^2 (b d-a e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{8 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^2 (b d-a e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^2 (b d-a e) (a+b x) \sqrt {d+e x}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (d+e x)^{5/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 185, normalized size = 0.73 \begin {gather*} \frac {e^2 (a+b x)^3 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (105 a^3 e^3+35 a^2 b e^2 (-4 d+5 e x)+7 a b^2 e \left (3 d^2-34 d e x+8 e^2 x^2\right )+b^3 \left (6 d^3+39 d^2 e x-80 d e^2 x^2-8 e^3 x^3\right )\right )}{e^2 (a+b x)^2}+105 (-b d+a e)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{12 b^{9/2} \left ((a+b x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(713\) vs.
\(2(171)=342\).
time = 0.73, size = 714, normalized size = 2.81
method | result | size |
risch | \(-\frac {2 e^{2} \left (-b e x +9 a e -10 b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{4} \left (b x +a \right )}+\frac {\left (-\frac {13 e^{4} \left (e x +d \right )^{\frac {3}{2}} a^{2}}{4 b^{3} \left (b e x +a e \right )^{2}}+\frac {13 e^{3} \left (e x +d \right )^{\frac {3}{2}} a d}{2 b^{2} \left (b e x +a e \right )^{2}}-\frac {13 e^{2} \left (e x +d \right )^{\frac {3}{2}} d^{2}}{4 b \left (b e x +a e \right )^{2}}-\frac {11 e^{5} \sqrt {e x +d}\, a^{3}}{4 b^{4} \left (b e x +a e \right )^{2}}+\frac {33 e^{4} \sqrt {e x +d}\, a^{2} d}{4 b^{3} \left (b e x +a e \right )^{2}}-\frac {33 e^{3} \sqrt {e x +d}\, a \,d^{2}}{4 b^{2} \left (b e x +a e \right )^{2}}+\frac {11 e^{2} \sqrt {e x +d}\, d^{3}}{4 b \left (b e x +a e \right )^{2}}+\frac {35 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2}}{4 b^{4} \sqrt {b \left (a e -b d \right )}}-\frac {35 e^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a d}{2 b^{3} \sqrt {b \left (a e -b d \right )}}+\frac {35 e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d^{2}}{4 b^{2} \sqrt {b \left (a e -b d \right )}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(396\) |
default | \(-\frac {\left (-8 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} e^{2} x^{2}-105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{2} e^{4} x^{2}+210 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{3} d \,e^{3} x^{2}-105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{4} d^{2} e^{2} x^{2}-16 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} e^{2} x +72 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} e^{3} x^{2}-72 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} d \,e^{2} x^{2}-210 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b \,e^{4} x +420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{2} d \,e^{3} x -210 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{3} d^{2} e^{2} x +31 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}-78 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e +39 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+144 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b \,e^{3} x -144 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} d \,e^{2} x -105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} e^{4}+210 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b d \,e^{3}-105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{2} d^{2} e^{2}+105 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{3} e^{3}-171 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b d \,e^{2}+99 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} d^{2} e -33 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} d^{3}\right ) \left (b x +a \right )}{12 \sqrt {b \left (a e -b d \right )}\, b^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(714\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.97, size = 489, normalized size = 1.93 \begin {gather*} \left [\frac {105 \, {\left ({\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} e^{3} - {\left (b^{3} d x^{2} + 2 \, a b^{2} d x + a^{2} b d\right )} e^{2}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d + 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (6 \, b^{3} d^{3} - {\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} e^{3} - 2 \, {\left (40 \, b^{3} d x^{2} + 119 \, a b^{2} d x + 70 \, a^{2} b d\right )} e^{2} + 3 \, {\left (13 \, b^{3} d^{2} x + 7 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {105 \, {\left ({\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} e^{3} - {\left (b^{3} d x^{2} + 2 \, a b^{2} d x + a^{2} b d\right )} e^{2}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (6 \, b^{3} d^{3} - {\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} e^{3} - 2 \, {\left (40 \, b^{3} d x^{2} + 119 \, a b^{2} d x + 70 \, a^{2} b d\right )} e^{2} + 3 \, {\left (13 \, b^{3} d^{2} x + 7 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.66, size = 289, normalized size = 1.14 \begin {gather*} \frac {35 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {13 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{2} - 11 \, \sqrt {x e + d} b^{3} d^{3} e^{2} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{3} + 33 \, \sqrt {x e + d} a b^{2} d^{2} e^{3} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{4} - 33 \, \sqrt {x e + d} a^{2} b d e^{4} + 11 \, \sqrt {x e + d} a^{3} e^{5}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{4} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{6} e^{2} + 9 \, \sqrt {x e + d} b^{6} d e^{2} - 9 \, \sqrt {x e + d} a b^{5} e^{3}\right )}}{3 \, b^{9} \mathrm {sgn}\left (b x + a\right )} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{7/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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