Integrand size = 24, antiderivative size = 246 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a+b x}} \, dx=-\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {5 (b d-a e)^3 (b B d-8 A b e+7 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{9/2} e^{3/2}} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {81, 52, 65, 223, 212} \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a+b x}} \, dx=-\frac {5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}-\frac {5 \sqrt {a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac {\sqrt {a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e} \]
[In]
[Out]
Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}+\frac {\left (4 A b e-B \left (\frac {b d}{2}+\frac {7 a e}{2}\right )\right ) \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}} \, dx}{4 b e} \\ & = -\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {(5 (b d-a e) (b B d-8 A b e+7 a B e)) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^2 e} \\ & = -\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {\left (5 (b d-a e)^2 (b B d-8 A b e+7 a B e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{64 b^3 e} \\ & = -\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {\left (5 (b d-a e)^3 (b B d-8 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{128 b^4 e} \\ & = -\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {\left (5 (b d-a e)^3 (b B d-8 A b e+7 a B e)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^5 e} \\ & = -\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {\left (5 (b d-a e)^3 (b B d-8 A b e+7 a B e)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{64 b^5 e} \\ & = -\frac {5 (b d-a e)^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^4 e}-\frac {5 (b d-a e) (b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{96 b^3 e}-\frac {(b B d-8 A b e+7 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{24 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{7/2}}{4 b e}-\frac {5 (b d-a e)^3 (b B d-8 A b e+7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{9/2} e^{3/2}} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (-105 a^3 B e^3+5 a^2 b e^2 (53 B d+24 A e+14 B e x)-a b^2 e \left (80 A e (4 d+e x)+B \left (191 d^2+172 d e x+56 e^2 x^2\right )\right )+b^3 \left (8 A e \left (33 d^2+26 d e x+8 e^2 x^2\right )+B \left (15 d^3+118 d^2 e x+136 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b^4 e}-\frac {5 (b d-a e)^3 (b B d-8 A b e+7 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{9/2} e^{3/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(208)=416\).
Time = 1.09 (sec) , antiderivative size = 968, normalized size of antiderivative = 3.93
method | result | size |
default | \(-\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (344 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d \,e^{2} x -105 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{4} e^{4}+15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{4}-528 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e -240 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3}+60 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{3} e -270 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d^{2} e^{2}-360 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d \,e^{3}+360 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{2} e^{2}+300 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b d \,e^{3}-530 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b d \,e^{2}+382 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d^{2} e +210 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{3} e^{3}-30 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{3}+120 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b \,e^{4}-120 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{3} e +160 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} e^{3} x -416 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d \,e^{2} x -140 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3} x -236 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e x +112 B a \,b^{2} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-272 B \,b^{3} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+640 A a \,b^{2} d \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-96 B \,b^{3} e^{3} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-128 A \,b^{3} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{384 b^{4} e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}\) | \(968\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.14 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (B b^{4} d^{4} + 4 \, {\left (B a b^{3} - 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 4 \, {\left (5 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (48 \, B b^{4} e^{4} x^{3} + 15 \, B b^{4} d^{3} e - {\left (191 \, B a b^{3} - 264 \, A b^{4}\right )} d^{2} e^{2} + 5 \, {\left (53 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d e^{3} - 15 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (17 \, B b^{4} d e^{3} - {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (59 \, B b^{4} d^{2} e^{2} - 2 \, {\left (43 \, B a b^{3} - 52 \, A b^{4}\right )} d e^{3} + 5 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{768 \, b^{5} e^{2}}, \frac {15 \, {\left (B b^{4} d^{4} + 4 \, {\left (B a b^{3} - 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 4 \, {\left (5 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, B b^{4} e^{4} x^{3} + 15 \, B b^{4} d^{3} e - {\left (191 \, B a b^{3} - 264 \, A b^{4}\right )} d^{2} e^{2} + 5 \, {\left (53 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d e^{3} - 15 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (17 \, B b^{4} d e^{3} - {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (59 \, B b^{4} d^{2} e^{2} - 2 \, {\left (43 \, B a b^{3} - 52 \, A b^{4}\right )} d e^{3} + 5 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{384 \, b^{5} e^{2}}\right ] \]
[In]
[Out]
\[ \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a+b x}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {5}{2}}}{\sqrt {a + b x}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1086 vs. \(2 (208) = 416\).
Time = 0.44 (sec) , antiderivative size = 1086, normalized size of antiderivative = 4.41 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a+b x}} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\sqrt {a+b x}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{\sqrt {a+b\,x}} \,d x \]
[In]
[Out]