Integrand size = 35, antiderivative size = 164 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 (b d-a e) (B d-A e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}+\frac {2 b B (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^3 (a+b x)} \]
[Out]
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} \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3 (a+b x)}+\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)} \]
[In]
[Out]
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)^{7/2} \, 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)^{7/2}}{e^2}+\frac {b (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^2}+\frac {b^2 B (d+e x)^{11/2}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 (b d-a e) (B d-A 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 (2 b B d-A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}+\frac {2 b B (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^3 (a+b x)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.54 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{9/2} \left (13 A b e (-2 d+9 e x)+13 a e (-2 B d+11 A e+9 B e x)+b B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3 (a+b x)} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {9}{2}} \left (99 B b \,e^{2} x^{2}+117 A b \,e^{2} x +117 B a \,e^{2} x -36 B b d e x +143 A a \,e^{2}-26 A b d e -26 B a d e +8 B b \,d^{2}\right )}{1287 e^{3}}\) | \(79\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 B b \,e^{2} x^{2}+117 A b \,e^{2} x +117 B a \,e^{2} x -36 B b d e x +143 A a \,e^{2}-26 A b d e -26 B a d e +8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{1287 e^{3} \left (b x +a \right )}\) | \(89\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (99 B \,e^{6} b \,x^{6}+117 A b \,e^{6} x^{5}+117 B a \,e^{6} x^{5}+360 B b d \,e^{5} x^{5}+143 A a \,e^{6} x^{4}+442 A b d \,e^{5} x^{4}+442 B a d \,e^{5} x^{4}+458 B b \,d^{2} e^{4} x^{4}+572 A a d \,e^{5} x^{3}+598 A b \,d^{2} e^{4} x^{3}+598 B a \,d^{2} e^{4} x^{3}+212 B b \,d^{3} e^{3} x^{3}+858 A a \,d^{2} e^{4} x^{2}+312 A b \,d^{3} e^{3} x^{2}+312 B a \,d^{3} e^{3} x^{2}+3 B b \,d^{4} e^{2} x^{2}+572 A a \,d^{3} e^{3} x +13 A b \,d^{4} e^{2} x +13 B a \,d^{4} e^{2} x -4 B b \,d^{5} e x +143 A a \,d^{4} e^{2}-26 A b \,d^{5} e -26 B a \,d^{5} e +8 B b \,d^{6}\right ) \sqrt {e x +d}}{1287 \left (b x +a \right ) e^{3}}\) | \(293\) |
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.40 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (99 \, B b e^{6} x^{6} + 8 \, B b d^{6} + 143 \, A a d^{4} e^{2} - 26 \, {\left (B a + A b\right )} d^{5} e + 9 \, {\left (40 \, B b d e^{5} + 13 \, {\left (B a + A b\right )} e^{6}\right )} x^{5} + {\left (458 \, B b d^{2} e^{4} + 143 \, A a e^{6} + 442 \, {\left (B a + A b\right )} d e^{5}\right )} x^{4} + 2 \, {\left (106 \, B b d^{3} e^{3} + 286 \, A a d e^{5} + 299 \, {\left (B a + A b\right )} d^{2} e^{4}\right )} x^{3} + 3 \, {\left (B b d^{4} e^{2} + 286 \, A a d^{2} e^{4} + 104 \, {\left (B a + A b\right )} d^{3} e^{3}\right )} x^{2} - {\left (4 \, B b d^{5} e - 572 \, A a d^{3} e^{3} - 13 \, {\left (B a + A b\right )} d^{4} e^{2}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \]
[In]
[Out]
Timed out. \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (119) = 238\).
Time = 0.22 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.60 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (9 \, b e^{5} x^{5} - 2 \, b d^{5} + 11 \, a d^{4} e + {\left (34 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + 2 \, {\left (23 \, b d^{2} e^{3} + 22 \, a d e^{4}\right )} x^{3} + 6 \, {\left (4 \, b d^{3} e^{2} + 11 \, a d^{2} e^{3}\right )} x^{2} + {\left (b d^{4} e + 44 \, a d^{3} e^{2}\right )} x\right )} \sqrt {e x + d} A}{99 \, e^{2}} + \frac {2 \, {\left (99 \, b e^{6} x^{6} + 8 \, b d^{6} - 26 \, a d^{5} e + 9 \, {\left (40 \, b d e^{5} + 13 \, a e^{6}\right )} x^{5} + 2 \, {\left (229 \, b d^{2} e^{4} + 221 \, a d e^{5}\right )} x^{4} + 2 \, {\left (106 \, b d^{3} e^{3} + 299 \, a d^{2} e^{4}\right )} x^{3} + 3 \, {\left (b d^{4} e^{2} + 104 \, a d^{3} e^{3}\right )} x^{2} - {\left (4 \, b d^{5} e - 13 \, a d^{4} e^{2}\right )} x\right )} \sqrt {e x + d} B}{1287 \, e^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1164 vs. \(2 (119) = 238\).
Time = 0.32 (sec) , antiderivative size = 1164, normalized size of antiderivative = 7.10 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \sqrt {{\left (a+b\,x\right )}^2}\,\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2} \,d x \]
[In]
[Out]