Integrand size = 35, antiderivative size = 164 \[ \int (A+B x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 (b d-a e) (B d-A e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}+\frac {2 b B (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)} \]
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Time = 0.06 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {784, 78} \[ \int (A+B x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3 (a+b x)}+\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^3 (a+b x)} \]
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Rule 78
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (A+B x) (d+e x)^{5/2} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (-B d+A e) (d+e x)^{5/2}}{e^2}+\frac {b (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^2}+\frac {b^2 B (d+e x)^{9/2}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 (b d-a e) (B d-A e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}+\frac {2 b B (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.54 \[ \int (A+B x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (11 A b e (-2 d+7 e x)+11 a e (-2 B d+9 A e+7 B e x)+b B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {7}{2}} \left (63 B b \,e^{2} x^{2}+77 A b \,e^{2} x +77 B a \,e^{2} x -28 B b d e x +99 A a \,e^{2}-22 A b d e -22 B a d e +8 B b \,d^{2}\right )}{693 e^{3}}\) | \(79\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 B b \,e^{2} x^{2}+77 A b \,e^{2} x +77 B a \,e^{2} x -28 B b d e x +99 A a \,e^{2}-22 A b d e -22 B a d e +8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{693 e^{3} \left (b x +a \right )}\) | \(89\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (63 B b \,e^{5} x^{5}+77 A b \,e^{5} x^{4}+77 B a \,e^{5} x^{4}+161 B b d \,e^{4} x^{4}+99 A a \,e^{5} x^{3}+209 A b d \,e^{4} x^{3}+209 B a d \,e^{4} x^{3}+113 B b \,d^{2} e^{3} x^{3}+297 A a d \,e^{4} x^{2}+165 A b \,d^{2} e^{3} x^{2}+165 B a \,d^{2} e^{3} x^{2}+3 B b \,d^{3} e^{2} x^{2}+297 A a \,d^{2} e^{3} x +11 A b \,d^{3} e^{2} x +11 B a \,d^{3} e^{2} x -4 B b \,d^{4} e x +99 A a \,d^{3} e^{2}-22 A b \,d^{4} e -22 B a \,d^{4} e +8 B b \,d^{5}\right ) \sqrt {e x +d}}{693 \left (b x +a \right ) e^{3}}\) | \(241\) |
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Time = 0.45 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.15 \[ \int (A+B x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (63 \, B b e^{5} x^{5} + 8 \, B b d^{5} + 99 \, A a d^{3} e^{2} - 22 \, {\left (B a + A b\right )} d^{4} e + 7 \, {\left (23 \, B b d e^{4} + 11 \, {\left (B a + A b\right )} e^{5}\right )} x^{4} + {\left (113 \, B b d^{2} e^{3} + 99 \, A a e^{5} + 209 \, {\left (B a + A b\right )} d e^{4}\right )} x^{3} + 3 \, {\left (B b d^{3} e^{2} + 99 \, A a d e^{4} + 55 \, {\left (B a + A b\right )} d^{2} e^{3}\right )} x^{2} - {\left (4 \, B b d^{4} e - 297 \, A a d^{2} e^{3} - 11 \, {\left (B a + A b\right )} d^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \]
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Timed out. \[ \int (A+B x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.30 \[ \int (A+B x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (7 \, b e^{4} x^{4} - 2 \, b d^{4} + 9 \, a d^{3} e + {\left (19 \, b d e^{3} + 9 \, a e^{4}\right )} x^{3} + 3 \, {\left (5 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{2} + {\left (b d^{3} e + 27 \, a d^{2} e^{2}\right )} x\right )} \sqrt {e x + d} A}{63 \, e^{2}} + \frac {2 \, {\left (63 \, b e^{5} x^{5} + 8 \, b d^{5} - 22 \, a d^{4} e + 7 \, {\left (23 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + {\left (113 \, b d^{2} e^{3} + 209 \, a d e^{4}\right )} x^{3} + 3 \, {\left (b d^{3} e^{2} + 55 \, a d^{2} e^{3}\right )} x^{2} - {\left (4 \, b d^{4} e - 11 \, a d^{3} e^{2}\right )} x\right )} \sqrt {e x + d} B}{693 \, e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (119) = 238\).
Time = 0.29 (sec) , antiderivative size = 831, normalized size of antiderivative = 5.07 \[ \int (A+B x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (A+B x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \sqrt {{\left (a+b\,x\right )}^2}\,\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2} \,d x \]
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