Optimal. Leaf size=169 \[ -\frac {2 (b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^{3/2}}+\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{e^5 \sqrt {d+e x}}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) \sqrt {d+e x}}{e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^3 B (d+e x)^{5/2}}{5 e^5} \]
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Rubi [A]
time = 0.04, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78}
\begin {gather*} -\frac {2 b^2 (d+e x)^{3/2} (-3 a B e-A b e+4 b B d)}{3 e^5}+\frac {6 b \sqrt {d+e x} (b d-a e) (-a B e-A b e+2 b B d)}{e^5}+\frac {2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 \sqrt {d+e x}}-\frac {2 (b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^{3/2}}+\frac {2 b^3 B (d+e x)^{5/2}}{5 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{5/2}}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{3/2}}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 \sqrt {d+e x}}+\frac {b^2 (-4 b B d+A b e+3 a B e) \sqrt {d+e x}}{e^4}+\frac {b^3 B (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac {2 (b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^{3/2}}+\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{e^5 \sqrt {d+e x}}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) \sqrt {d+e x}}{e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^3 B (d+e x)^{5/2}}{5 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 223, normalized size = 1.32 \begin {gather*} \frac {2 \left (-5 a^3 e^3 (2 B d+A e+3 B e x)+15 a^2 b e^2 \left (-A e (2 d+3 e x)+B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+15 a b^2 e \left (A e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )\right )+b^3 \left (5 A e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+B \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )\right )}{15 e^5 (d+e x)^{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. \(316\) vs.
\(2(153)=306\).
time = 0.09, size = 317, normalized size = 1.88
method | result | size |
risch | \(\frac {2 b \left (3 b^{2} B \,x^{2} e^{2}+5 A \,b^{2} e^{2} x +15 B a b \,e^{2} x -14 B \,b^{2} d e x +45 A a b \,e^{2}-40 A \,b^{2} d e +45 B \,a^{2} e^{2}-120 B a b d e +73 b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{15 e^{5}}-\frac {2 \left (9 A b \,e^{2} x +3 B a \,e^{2} x -12 B b d e x +A a \,e^{2}+8 A b d e +2 B a d e -11 B b \,d^{2}\right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\) | \(179\) |
gosper | \(-\frac {2 \left (-3 b^{3} B \,x^{4} e^{4}-5 A \,b^{3} e^{4} x^{3}-15 B a \,b^{2} e^{4} x^{3}+8 B \,b^{3} d \,e^{3} x^{3}-45 A a \,b^{2} e^{4} x^{2}+30 A \,b^{3} d \,e^{3} x^{2}-45 B \,a^{2} b \,e^{4} x^{2}+90 B a \,b^{2} d \,e^{3} x^{2}-48 B \,b^{3} d^{2} e^{2} x^{2}+45 A \,a^{2} b \,e^{4} x -180 A a \,b^{2} d \,e^{3} x +120 A \,b^{3} d^{2} e^{2} x +15 B \,a^{3} e^{4} x -180 B \,a^{2} b d \,e^{3} x +360 B a \,b^{2} d^{2} e^{2} x -192 B \,b^{3} d^{3} e x +5 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}-120 A a \,b^{2} d^{2} e^{2}+80 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}-120 B \,a^{2} b \,d^{2} e^{2}+240 B a \,b^{2} d^{3} e -128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\) | \(301\) |
trager | \(-\frac {2 \left (-3 b^{3} B \,x^{4} e^{4}-5 A \,b^{3} e^{4} x^{3}-15 B a \,b^{2} e^{4} x^{3}+8 B \,b^{3} d \,e^{3} x^{3}-45 A a \,b^{2} e^{4} x^{2}+30 A \,b^{3} d \,e^{3} x^{2}-45 B \,a^{2} b \,e^{4} x^{2}+90 B a \,b^{2} d \,e^{3} x^{2}-48 B \,b^{3} d^{2} e^{2} x^{2}+45 A \,a^{2} b \,e^{4} x -180 A a \,b^{2} d \,e^{3} x +120 A \,b^{3} d^{2} e^{2} x +15 B \,a^{3} e^{4} x -180 B \,a^{2} b d \,e^{3} x +360 B a \,b^{2} d^{2} e^{2} x -192 B \,b^{3} d^{3} e x +5 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}-120 A a \,b^{2} d^{2} e^{2}+80 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}-120 B \,a^{2} b \,d^{2} e^{2}+240 B a \,b^{2} d^{3} e -128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\) | \(301\) |
derivativedivides | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B a \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {8 B \,b^{3} d \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A a \,b^{2} e^{2} \sqrt {e x +d}-6 A \,b^{3} d e \sqrt {e x +d}+6 B \,a^{2} b \,e^{2} \sqrt {e x +d}-18 B a \,b^{2} d e \sqrt {e x +d}+12 B \,b^{3} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(317\) |
default | \(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B a \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {8 B \,b^{3} d \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A a \,b^{2} e^{2} \sqrt {e x +d}-6 A \,b^{3} d e \sqrt {e x +d}+6 B \,a^{2} b \,e^{2} \sqrt {e x +d}-18 B a \,b^{2} d e \sqrt {e x +d}+12 B \,b^{3} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(317\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 283, normalized size = 1.67 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} - 5 \, {\left (4 \, B b^{3} d - 3 \, B a b^{2} e - A b^{3} e\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 45 \, {\left (2 \, B b^{3} d^{2} + B a^{2} b e^{2} + A a b^{2} e^{2} - {\left (3 \, B a b^{2} e + A b^{3} e\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {5 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{3} + 3 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d^{2} - 3 \, {\left (4 \, B b^{3} d^{3} - B a^{3} e^{3} - 3 \, A a^{2} b e^{3} - 3 \, {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{2} + 6 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d\right )} {\left (x e + d\right )} - {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 266, normalized size = 1.57 \begin {gather*} \frac {2 \, {\left (128 \, B b^{3} d^{4} + {\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{4} - 2 \, {\left (4 \, B b^{3} d x^{3} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{2} - 90 \, {\left (B a^{2} b + A a b^{2}\right )} d x + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{3} + 24 \, {\left (2 \, B b^{3} d^{2} x^{2} - 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x + 5 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2}\right )} e^{2} + 16 \, {\left (12 \, B b^{3} d^{3} x - 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 35.01, size = 199, normalized size = 1.18 \begin {gather*} \frac {2 B b^{3} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 A b^{3} e + 6 B a b^{2} e - 8 B b^{3} d\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (6 A a b^{2} e^{2} - 6 A b^{3} d e + 6 B a^{2} b e^{2} - 18 B a b^{2} d e + 12 B b^{3} d^{2}\right )}{e^{5}} - \frac {2 \left (a e - b d\right )^{2} \cdot \left (3 A b e + B a e - 4 B b d\right )}{e^{5} \sqrt {d + e x}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{3}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs.
\(2 (163) = 326\).
time = 0.81, size = 365, normalized size = 2.16 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} e^{20} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d e^{20} + 90 \, \sqrt {x e + d} B b^{3} d^{2} e^{20} + 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} e^{21} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} e^{21} - 135 \, \sqrt {x e + d} B a b^{2} d e^{21} - 45 \, \sqrt {x e + d} A b^{3} d e^{21} + 45 \, \sqrt {x e + d} B a^{2} b e^{22} + 45 \, \sqrt {x e + d} A a b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac {2 \, {\left (12 \, {\left (x e + d\right )} B b^{3} d^{3} - B b^{3} d^{4} - 27 \, {\left (x e + d\right )} B a b^{2} d^{2} e - 9 \, {\left (x e + d\right )} A b^{3} d^{2} e + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 18 \, {\left (x e + d\right )} B a^{2} b d e^{2} + 18 \, {\left (x e + d\right )} A a b^{2} d e^{2} - 3 \, B a^{2} b d^{2} e^{2} - 3 \, A a b^{2} d^{2} e^{2} - 3 \, {\left (x e + d\right )} B a^{3} e^{3} - 9 \, {\left (x e + d\right )} A a^{2} b e^{3} + B a^{3} d e^{3} + 3 \, A a^{2} b d e^{3} - A a^{3} e^{4}\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.27, size = 264, normalized size = 1.56 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{3\,e^5}-\frac {\left (d+e\,x\right )\,\left (2\,B\,a^3\,e^3-12\,B\,a^2\,b\,d\,e^2+6\,A\,a^2\,b\,e^3+18\,B\,a\,b^2\,d^2\,e-12\,A\,a\,b^2\,d\,e^2-8\,B\,b^3\,d^3+6\,A\,b^3\,d^2\,e\right )+\frac {2\,A\,a^3\,e^4}{3}+\frac {2\,B\,b^3\,d^4}{3}-\frac {2\,A\,b^3\,d^3\,e}{3}-\frac {2\,B\,a^3\,d\,e^3}{3}+2\,A\,a\,b^2\,d^2\,e^2+2\,B\,a^2\,b\,d^2\,e^2-2\,A\,a^2\,b\,d\,e^3-2\,B\,a\,b^2\,d^3\,e}{e^5\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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