Optimal. Leaf size=124 \[ \frac {2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt {d+e x}}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) \sqrt {d+e x}}{e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{3/2}}{3 e^4}+\frac {2 b^2 B (d+e x)^{5/2}}{5 e^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 124, 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 (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac {2 \sqrt {d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac {2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt {d+e x}}+\frac {2 b^2 B (d+e x)^{5/2}}{5 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^{3/2}}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 \sqrt {d+e x}}+\frac {b (-3 b B d+A b e+2 a B e) \sqrt {d+e x}}{e^3}+\frac {b^2 B (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac {2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt {d+e x}}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) \sqrt {d+e x}}{e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{3/2}}{3 e^4}+\frac {2 b^2 B (d+e x)^{5/2}}{5 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 135, normalized size = 1.09 \begin {gather*} \frac {30 a^2 e^2 (2 B d-A e+B e x)+20 a b e \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+2 b^2 \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 200, normalized size = 1.61
method | result | size |
risch | \(\frac {2 \left (3 b^{2} B \,x^{2} e^{2}+5 b^{2} A x \,e^{2}+10 a b B x \,e^{2}-9 B \,b^{2} d e x +30 A a b \,e^{2}-25 A \,b^{2} d e +15 B \,a^{2} e^{2}-50 B a b d e +33 b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{15 e^{4}}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{e^{4} \sqrt {e x +d}}\) | \(163\) |
gosper | \(-\frac {2 \left (-3 b^{2} B \,x^{3} e^{3}-5 A \,b^{2} e^{3} x^{2}-10 B a b \,e^{3} x^{2}+6 B \,b^{2} d \,e^{2} x^{2}-30 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x -15 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -24 B \,b^{2} d^{2} e x +15 a^{2} A \,e^{3}-60 A a b d \,e^{2}+40 A \,b^{2} d^{2} e -30 B \,a^{2} d \,e^{2}+80 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{15 \sqrt {e x +d}\, e^{4}}\) | \(169\) |
trager | \(-\frac {2 \left (-3 b^{2} B \,x^{3} e^{3}-5 A \,b^{2} e^{3} x^{2}-10 B a b \,e^{3} x^{2}+6 B \,b^{2} d \,e^{2} x^{2}-30 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x -15 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -24 B \,b^{2} d^{2} e x +15 a^{2} A \,e^{3}-60 A a b d \,e^{2}+40 A \,b^{2} d^{2} e -30 B \,a^{2} d \,e^{2}+80 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{15 \sqrt {e x +d}\, e^{4}}\) | \(169\) |
derivativedivides | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 B a b e \left (e x +d \right )^{\frac {3}{2}}}{3}-2 B \,b^{2} d \left (e x +d \right )^{\frac {3}{2}}+4 A a b \,e^{2} \sqrt {e x +d}-4 A \,b^{2} d e \sqrt {e x +d}+2 B \,a^{2} e^{2} \sqrt {e x +d}-8 B a b d e \sqrt {e x +d}+6 B \,b^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{\sqrt {e x +d}}}{e^{4}}\) | \(200\) |
default | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 B a b e \left (e x +d \right )^{\frac {3}{2}}}{3}-2 B \,b^{2} d \left (e x +d \right )^{\frac {3}{2}}+4 A a b \,e^{2} \sqrt {e x +d}-4 A \,b^{2} d e \sqrt {e x +d}+2 B \,a^{2} e^{2} \sqrt {e x +d}-8 B a b d e \sqrt {e x +d}+6 B \,b^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{\sqrt {e x +d}}}{e^{4}}\) | \(200\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.45, size = 174, normalized size = 1.40 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{2} - 5 \, {\left (3 \, B b^{2} d - 2 \, B a b e - A b^{2} e\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 15 \, {\left (3 \, B b^{2} d^{2} + B a^{2} e^{2} + 2 \, A a b e^{2} - 2 \, {\left (2 \, B a b e + A b^{2} e\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-3\right )} + \frac {15 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b e + A b^{2} e\right )} d^{2} + {\left (B a^{2} e^{2} + 2 \, A a b e^{2}\right )} d\right )} e^{\left (-3\right )}}{\sqrt {x e + d}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.00, size = 155, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (48 \, B b^{2} d^{3} + {\left (3 \, B b^{2} x^{3} - 15 \, A a^{2} + 5 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )} e^{3} - 2 \, {\left (3 \, B b^{2} d x^{2} + 10 \, {\left (2 \, B a b + A b^{2}\right )} d x - 15 \, {\left (B a^{2} + 2 \, A a b\right )} d\right )} e^{2} + 8 \, {\left (3 \, B b^{2} d^{2} x - 5 \, {\left (2 \, B a b + A b^{2}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x e^{5} + d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.31, size = 150, normalized size = 1.21 \begin {gather*} \frac {2 B b^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 A b^{2} e + 4 B a b e - 6 B b^{2} d\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (4 A a b e^{2} - 4 A b^{2} d e + 2 B a^{2} e^{2} - 8 B a b d e + 6 B b^{2} d^{2}\right )}{e^{4}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{2}}{e^{4} \sqrt {d + e x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 219, normalized size = 1.77 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{2} e^{16} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} d e^{16} + 45 \, \sqrt {x e + d} B b^{2} d^{2} e^{16} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b e^{17} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} e^{17} - 60 \, \sqrt {x e + d} B a b d e^{17} - 30 \, \sqrt {x e + d} A b^{2} d e^{17} + 15 \, \sqrt {x e + d} B a^{2} e^{18} + 30 \, \sqrt {x e + d} A a b e^{18}\right )} e^{\left (-20\right )} + \frac {2 \, {\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} e^{\left (-4\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 154, normalized size = 1.24 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{3\,e^4}-\frac {-2\,B\,a^2\,d\,e^2+2\,A\,a^2\,e^3+4\,B\,a\,b\,d^2\,e-4\,A\,a\,b\,d\,e^2-2\,B\,b^2\,d^3+2\,A\,b^2\,d^2\,e}{e^4\,\sqrt {d+e\,x}}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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