Optimal. Leaf size=124 \[ \frac {2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}-\frac {2 (b d-a e) (3 b B d-2 A b e-a B e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 b (3 b B d-A b e-2 a B e)}{e^4 \sqrt {d+e x}}+\frac {2 b^2 B \sqrt {d+e x}}{e^4} \]
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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 (-2 a B e-A b e+3 b B d)}{e^4 \sqrt {d+e x}}-\frac {2 (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4 (d+e x)^{3/2}}+\frac {2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac {2 b^2 B \sqrt {d+e x}}{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)^{7/2}} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^{7/2}}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^{5/2}}+\frac {b (-3 b B d+A b e+2 a B e)}{e^3 (d+e x)^{3/2}}+\frac {b^2 B}{e^3 \sqrt {d+e x}}\right ) \, dx\\ &=\frac {2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}-\frac {2 (b d-a e) (3 b B d-2 A b e-a B e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 b (3 b B d-A b e-2 a B e)}{e^4 \sqrt {d+e x}}+\frac {2 b^2 B \sqrt {d+e x}}{e^4}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 136, normalized size = 1.10 \begin {gather*} -\frac {2 \left (a^2 e^2 (2 B d+3 A e+5 B e x)+2 a b e \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+b^2 \left (A e \left (8 d^2+20 d e x+15 e^2 x^2\right )-3 B \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{15 e^4 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 158, normalized size = 1.27
method | result | size |
derivativedivides | \(\frac {2 b^{2} B \sqrt {e x +d}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +B \,a^{2} e^{2}-4 B a b d e +3 b^{2} B \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 b \left (A b e +2 B a e -3 B b d \right )}{\sqrt {e x +d}}}{e^{4}}\) | \(158\) |
default | \(\frac {2 b^{2} B \sqrt {e x +d}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +B \,a^{2} e^{2}-4 B a b d e +3 b^{2} B \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 b \left (A b e +2 B a e -3 B b d \right )}{\sqrt {e x +d}}}{e^{4}}\) | \(158\) |
gosper | \(-\frac {2 \left (-15 b^{2} B \,x^{3} e^{3}+15 A \,b^{2} e^{3} x^{2}+30 B a b \,e^{3} x^{2}-90 B \,b^{2} d \,e^{2} x^{2}+10 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x +5 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -120 B \,b^{2} d^{2} e x +3 a^{2} A \,e^{3}+4 A a b d \,e^{2}+8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{4}}\) | \(169\) |
trager | \(-\frac {2 \left (-15 b^{2} B \,x^{3} e^{3}+15 A \,b^{2} e^{3} x^{2}+30 B a b \,e^{3} x^{2}-90 B \,b^{2} d \,e^{2} x^{2}+10 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x +5 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -120 B \,b^{2} d^{2} e x +3 a^{2} A \,e^{3}+4 A a b d \,e^{2}+8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{4}}\) | \(169\) |
risch | \(\frac {2 b^{2} B \sqrt {e x +d}}{e^{4}}-\frac {2 \left (15 A \,b^{2} e^{3} x^{2}+30 B a b \,e^{3} x^{2}-45 B \,b^{2} d \,e^{2} x^{2}+10 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x +5 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -75 B \,b^{2} d^{2} e x +3 a^{2} A \,e^{3}+4 A a b d \,e^{2}+8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -33 b^{2} B \,d^{3}\right )}{15 e^{4} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 171, normalized size = 1.38 \begin {gather*} \frac {2}{15} \, {\left (15 \, \sqrt {x e + d} B b^{2} e^{\left (-3\right )} + \frac {{\left (3 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} + 15 \, {\left (3 \, B b^{2} d - 2 \, B a b e - A b^{2} e\right )} {\left (x e + d\right )}^{2} - 3 \, {\left (2 \, B a b e + A b^{2} e\right )} d^{2} - 5 \, {\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 )} {\left (x e + d\right )} + 3 \, {\left (B a^{2} e^{2} + 2 \, A a b e^{2}\right )} d\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{\frac {5}{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 = 0.71, size = 175, normalized size = 1.41 \begin {gather*} \frac {2 \, {\left (48 \, B b^{2} d^{3} + {\left (15 \, B b^{2} x^{3} - 3 \, A a^{2} - 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} - 5 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )} e^{3} + 2 \, {\left (45 \, B b^{2} d x^{2} - 10 \, {\left (2 \, B a b + A b^{2}\right )} d x - {\left (B a^{2} + 2 \, A a b\right )} d\right )} e^{2} + 8 \, {\left (15 \, B b^{2} d^{2} x - {\left (2 \, B a b + A b^{2}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + d^{3} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1015 vs.
\(2 (126) = 252\).
time = 0.73, size = 1015, normalized size = 8.19 \begin {gather*} \begin {cases} - \frac {6 A a^{2} e^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {8 A a b d e^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {20 A a b e^{3} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {16 A b^{2} d^{2} e}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {40 A b^{2} d e^{2} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {30 A b^{2} e^{3} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {4 B a^{2} d e^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {10 B a^{2} e^{3} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {32 B a b d^{2} e}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {80 B a b d e^{2} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {60 B a b e^{3} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {96 B b^{2} d^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {240 B b^{2} d^{2} e x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {180 B b^{2} d e^{2} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {30 B b^{2} e^{3} x^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 202, normalized size = 1.63 \begin {gather*} 2 \, \sqrt {x e + d} B b^{2} e^{\left (-4\right )} + \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} B b^{2} d - 15 \, {\left (x e + d\right )} B b^{2} d^{2} + 3 \, B b^{2} d^{3} - 30 \, {\left (x e + d\right )}^{2} B a b e - 15 \, {\left (x e + d\right )}^{2} A b^{2} e + 20 \, {\left (x e + d\right )} B a b d e + 10 \, {\left (x e + d\right )} A b^{2} d e - 6 \, B a b d^{2} e - 3 \, A b^{2} d^{2} e - 5 \, {\left (x e + d\right )} B a^{2} e^{2} - 10 \, {\left (x e + d\right )} A a b e^{2} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} - 3 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 168, normalized size = 1.35 \begin {gather*} -\frac {2\,\left (2\,B\,a^2\,d\,e^2+5\,B\,a^2\,e^3\,x+3\,A\,a^2\,e^3+16\,B\,a\,b\,d^2\,e+40\,B\,a\,b\,d\,e^2\,x+4\,A\,a\,b\,d\,e^2+30\,B\,a\,b\,e^3\,x^2+10\,A\,a\,b\,e^3\,x-48\,B\,b^2\,d^3-120\,B\,b^2\,d^2\,e\,x+8\,A\,b^2\,d^2\,e-90\,B\,b^2\,d\,e^2\,x^2+20\,A\,b^2\,d\,e^2\,x-15\,B\,b^2\,e^3\,x^3+15\,A\,b^2\,e^3\,x^2\right )}{15\,e^4\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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