Integrand size = 35, antiderivative size = 489 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 \sqrt {b d-a e} (8 b B d+3 A b e-11 a B e) (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.27 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {784, 79, 43, 52, 65, 214} \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {105 e^3 (a+b x) \sqrt {b d-a e} (-11 a B e+3 A b e+8 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {105 e^3 (a+b x) \sqrt {d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 214
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (b^2 d-a b e\right ) (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (b^2 d-a b e\right )^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^2 \left (b^2 d-a b e\right )^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 \sqrt {b d-a e} (8 b B d+3 A b e-11 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 2.28 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3 (a+b x) \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3 A b \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )+B \left (3465 a^5 e^4+105 a^4 b e^3 (-35 d+121 e x)+21 a^3 b^2 e^2 \left (18 d^2-649 d e x+803 e^2 x^2\right )+9 a^2 b^3 e \left (8 d^3+164 d^2 e x-2041 d e^2 x^2+1023 e^3 x^3\right )+8 b^5 x \left (8 d^4+50 d^3 e x+165 d^2 e^2 x^2-208 d e^3 x^3-16 e^4 x^4\right )+a b^4 \left (16 d^4+280 d^3 e x+2130 d^2 e^2 x^2-10271 d e^3 x^3+1408 e^4 x^4\right )\right )\right )}{e^3 (a+b x)^4}-315 \sqrt {-b d+a e} (8 b B d+3 A b e-11 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{192 b^{13/2} \sqrt {(a+b x)^2}} \]
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Time = 0.33 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {2 e^{3} \left (B b e x +3 A b e -15 B a e +13 B b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{6} \left (b x +a \right )}-\frac {\left (2 a e -2 b d \right ) e^{3} \left (\frac {\left (-\frac {325}{128} A \,b^{4} e +\frac {765}{128} B e \,b^{3} a -\frac {55}{16} B \,b^{4} d \right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {5 b^{2} \left (459 A a b \,e^{2}-459 A \,b^{2} d e -1171 a^{2} B \,e^{2}+1883 B a b d e -712 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {643}{128} A \,a^{2} b^{2} e^{3}+\frac {643}{64} A a \,b^{3} d \,e^{2}-\frac {643}{128} A \,b^{4} d^{2} e +\frac {5153}{384} B \,e^{3} b \,a^{3}-\frac {2255}{64} B \,a^{2} b^{2} d \,e^{2}+\frac {3867}{128} B a \,b^{3} d^{2} e -\frac {403}{48} B \,b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} A \,a^{3} b \,e^{4}+\frac {561}{128} A \,a^{2} b^{2} d \,e^{3}-\frac {561}{128} A a \,b^{3} d^{2} e^{2}+\frac {187}{128} A \,b^{4} d^{3} e +\frac {515}{128} B \,a^{4} e^{4}-\frac {1873}{128} B \,a^{3} b d \,e^{3}+\frac {2529}{128} B \,a^{2} b^{2} d^{2} e^{2}-\frac {1499}{128} B a \,b^{3} d^{3} e +\frac {41}{16} b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {105 \left (3 A b e -11 B a e +8 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{6} \left (b x +a \right )}\) | \(439\) |
default | \(\text {Expression too large to display}\) | \(2430\) |
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Time = 0.50 (sec) , antiderivative size = 1414, normalized size of antiderivative = 2.89 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 824 vs. \(2 (376) = 752\).
Time = 0.32 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.69 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {105 \, {\left (8 \, B b^{2} d^{2} e^{3} - 19 \, B a b d e^{4} + 3 \, A b^{2} d e^{4} + 11 \, B a^{2} e^{5} - 3 \, A a b e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {1320 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{3} - 3560 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{3} + 3224 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{3} - 984 \, \sqrt {e x + d} B b^{5} d^{5} e^{3} - 3615 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{4} d e^{4} + 975 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{5} d e^{4} + 12975 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{4} - 2295 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{4} - 14825 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{4} + 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{4} + 5481 \, \sqrt {e x + d} B a b^{4} d^{4} e^{4} - 561 \, \sqrt {e x + d} A b^{5} d^{4} e^{4} + 2295 \, {\left (e x + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{5} - 975 \, {\left (e x + d\right )}^{\frac {7}{2}} A a b^{4} e^{5} - 15270 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{5} + 4590 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{4} d e^{5} + 25131 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{5} - 5787 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{5} - 12084 \, \sqrt {e x + d} B a^{2} b^{3} d^{3} e^{5} + 2244 \, \sqrt {e x + d} A a b^{4} d^{3} e^{5} + 5855 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{6} - 2295 \, {\left (e x + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{6} - 18683 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{6} + 5787 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{6} + 13206 \, \sqrt {e x + d} B a^{3} b^{2} d^{2} e^{6} - 3366 \, \sqrt {e x + d} A a^{2} b^{3} d^{2} e^{6} + 5153 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{4} b e^{7} - 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{7} - 7164 \, \sqrt {e x + d} B a^{4} b d e^{7} + 2244 \, \sqrt {e x + d} A a^{3} b^{2} d e^{7} + 1545 \, \sqrt {e x + d} B a^{5} e^{8} - 561 \, \sqrt {e x + d} A a^{4} b e^{8}}{192 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{10} e^{3} + 12 \, \sqrt {e x + d} B b^{10} d e^{3} - 15 \, \sqrt {e x + d} B a b^{9} e^{4} + 3 \, \sqrt {e x + d} A b^{10} e^{4}\right )}}{3 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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