Optimal. Leaf size=95 \[ -\frac {2 (B d-A e) (a+b x)^{7/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (7 b B d+2 A b e-9 a B e) (a+b x)^{7/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {79, 37}
\begin {gather*} \frac {2 (a+b x)^{7/2} (-9 a B e+2 A b e+7 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {(7 b B d+2 A b e-9 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{9 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (7 b B d+2 A b e-9 a B e) (a+b x)^{7/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 66, normalized size = 0.69 \begin {gather*} \frac {2 (a+b x)^{7/2} (B (-2 a d+7 b d x-9 a e x)+A (9 b d-7 a e+2 b e x))}{63 (b d-a e)^2 (d+e x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 159, normalized size = 1.67
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-2 A b e x +9 B a e x -7 B b d x +7 A a e -9 A b d +2 B a d \right )}{63 \left (e x +d \right )^{\frac {9}{2}} \left (a^{2} e^{2}-2 b e a d +b^{2} d^{2}\right )}\) | \(74\) |
default | \(-\frac {2 \left (-2 A \,b^{3} e \,x^{3}+9 B a \,b^{2} e \,x^{3}-7 B \,b^{3} d \,x^{3}+3 A a \,b^{2} e \,x^{2}-9 A \,b^{3} d \,x^{2}+18 B \,a^{2} b e \,x^{2}-12 B a \,b^{2} d \,x^{2}+12 A \,a^{2} b e x -18 A a \,b^{2} d x +9 B \,a^{3} e x -3 B \,a^{2} b d x +7 a^{3} A e -9 A \,a^{2} b d +2 B \,a^{3} d \right ) \left (b x +a \right )^{\frac {3}{2}}}{63 \left (e x +d \right )^{\frac {9}{2}} \left (a e -b d \right )^{2}}\) | \(159\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 389 vs.
\(2 (88) = 176\).
time = 37.14, size = 389, normalized size = 4.09 \begin {gather*} \frac {2 \, {\left (7 \, B b^{4} d x^{4} + {\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} d x^{3} + 3 \, {\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} d x^{2} + {\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} d x - {\left (2 \, B a^{4} - 9 \, A a^{3} b\right )} d - {\left (7 \, A a^{4} + {\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} + {\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} x^{3} + 3 \, {\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{2} + {\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} x\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{63 \, {\left (b^{2} d^{7} + a^{2} x^{5} e^{7} - {\left (2 \, a b d x^{5} - 5 \, a^{2} d x^{4}\right )} e^{6} + {\left (b^{2} d^{2} x^{5} - 10 \, a b d^{2} x^{4} + 10 \, a^{2} d^{2} x^{3}\right )} e^{5} + 5 \, {\left (b^{2} d^{3} x^{4} - 4 \, a b d^{3} x^{3} + 2 \, a^{2} d^{3} x^{2}\right )} e^{4} + 5 \, {\left (2 \, b^{2} d^{4} x^{3} - 4 \, a b d^{4} x^{2} + a^{2} d^{4} x\right )} e^{3} + {\left (10 \, b^{2} d^{5} x^{2} - 10 \, a b d^{5} x + a^{2} d^{5}\right )} e^{2} + {\left (5 \, b^{2} d^{6} x - 2 \, a b d^{6}\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs.
\(2 (88) = 176\).
time = 1.43, size = 356, normalized size = 3.75 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} {\left (\frac {{\left (7 \, B b^{12} d^{3} {\left | b \right |} e^{4} - 23 \, B a b^{11} d^{2} {\left | b \right |} e^{5} + 2 \, A b^{12} d^{2} {\left | b \right |} e^{5} + 25 \, B a^{2} b^{10} d {\left | b \right |} e^{6} - 4 \, A a b^{11} d {\left | b \right |} e^{6} - 9 \, B a^{3} b^{9} {\left | b \right |} e^{7} + 2 \, A a^{2} b^{10} {\left | b \right |} e^{7}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}} - \frac {9 \, {\left (B a b^{12} d^{3} {\left | b \right |} e^{4} - A b^{13} d^{3} {\left | b \right |} e^{4} - 3 \, B a^{2} b^{11} d^{2} {\left | b \right |} e^{5} + 3 \, A a b^{12} d^{2} {\left | b \right |} e^{5} + 3 \, B a^{3} b^{10} d {\left | b \right |} e^{6} - 3 \, A a^{2} b^{11} d {\left | b \right |} e^{6} - B a^{4} b^{9} {\left | b \right |} e^{7} + A a^{3} b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )}}{63 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.40, size = 325, normalized size = 3.42 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {x^3\,\sqrt {a+b\,x}\,\left (18\,A\,b^4\,d-2\,A\,a\,b^3\,e+38\,B\,a\,b^3\,d-54\,B\,a^2\,b^2\,e\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}-\frac {\sqrt {a+b\,x}\,\left (14\,A\,a^4\,e+4\,B\,a^4\,d-18\,A\,a^3\,b\,d\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}+\frac {x^4\,\sqrt {a+b\,x}\,\left (4\,A\,b^4\,e+14\,B\,b^4\,d-18\,B\,a\,b^3\,e\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}-\frac {x\,\sqrt {a+b\,x}\,\left (18\,B\,a^4\,e+38\,A\,a^3\,b\,e-2\,B\,a^3\,b\,d-54\,A\,a^2\,b^2\,d\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}+\frac {2\,a\,b\,x^2\,\sqrt {a+b\,x}\,\left (9\,A\,b^2\,d-9\,B\,a^2\,e-5\,A\,a\,b\,e+5\,B\,a\,b\,d\right )}{21\,e^5\,{\left (a\,e-b\,d\right )}^2}\right )}{x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________