Optimal. Leaf size=147 \[ -\frac {2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {4 b (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37}
\begin {gather*} \frac {4 b (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {(7 b B d+4 A b e-11 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{11 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {(2 b (7 b B d+4 A b e-11 a B e)) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{99 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {4 b (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 134, normalized size = 0.91 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (-63 B d e+63 A e^2+\frac {77 b B d (d+e x)}{a+b x}-\frac {154 A b e (d+e x)}{a+b x}+\frac {77 a B e (d+e x)}{a+b x}+\frac {99 A b^2 (d+e x)^2}{(a+b x)^2}-\frac {99 a b B (d+e x)^2}{(a+b x)^2}\right )}{693 (b d-a e)^3 (d+e x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(320\) vs.
\(2(129)=258\).
time = 0.13, size = 321, normalized size = 2.18
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (8 A \,b^{2} e^{2} x^{2}-22 B a b \,e^{2} x^{2}+14 B \,b^{2} d e \,x^{2}-28 A a b \,e^{2} x +44 A \,b^{2} d e x +77 B \,a^{2} e^{2} x -170 B a b d e x +77 B \,b^{2} d^{2} x +63 a^{2} A \,e^{2}-154 A a b d e +99 A \,b^{2} d^{2}+14 B \,a^{2} d e -22 B a b \,d^{2}\right )}{693 \left (e x +d \right )^{\frac {11}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(177\) |
default | \(-\frac {2 \left (8 A \,b^{4} e^{2} x^{4}-22 B a \,b^{3} e^{2} x^{4}+14 B \,b^{4} d e \,x^{4}-12 A a \,b^{3} e^{2} x^{3}+44 A \,b^{4} d e \,x^{3}+33 B \,a^{2} b^{2} e^{2} x^{3}-142 B a \,b^{3} d e \,x^{3}+77 B \,b^{4} d^{2} x^{3}+15 A \,a^{2} b^{2} e^{2} x^{2}-66 A a \,b^{3} d e \,x^{2}+99 A \,b^{4} d^{2} x^{2}+132 B \,a^{3} b \,e^{2} x^{2}-312 B \,a^{2} b^{2} d e \,x^{2}+132 B a \,b^{3} d^{2} x^{2}+98 A \,a^{3} b \,e^{2} x -264 A \,a^{2} b^{2} d e x +198 A a \,b^{3} d^{2} x +77 B \,a^{4} e^{2} x -142 B \,a^{3} b d e x +33 B \,a^{2} b^{2} d^{2} x +63 A \,a^{4} e^{2}-154 A \,a^{3} b d e +99 A \,a^{2} b^{2} d^{2}+14 B \,a^{4} d e -22 B \,a^{3} b \,d^{2}\right ) \left (b x +a \right )^{\frac {3}{2}}}{693 \left (e x +d \right )^{\frac {11}{2}} \left (a e -b d \right )^{3}}\) | \(321\) |
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. 687 vs.
\(2 (137) = 274\).
time = 95.31, size = 687, normalized size = 4.67 \begin {gather*} \frac {2 \, {\left (77 \, B b^{5} d^{2} x^{4} + 11 \, {\left (19 \, B a b^{4} + 9 \, A b^{5}\right )} d^{2} x^{3} + 33 \, {\left (5 \, B a^{2} b^{3} + 9 \, A a b^{4}\right )} d^{2} x^{2} + 11 \, {\left (B a^{3} b^{2} + 27 \, A a^{2} b^{3}\right )} d^{2} x - 11 \, {\left (2 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} d^{2} + {\left (63 \, A a^{5} - 2 \, {\left (11 \, B a b^{4} - 4 \, A b^{5}\right )} x^{5} + {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{4} + 3 \, {\left (55 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + {\left (209 \, B a^{4} b + 113 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (11 \, B a^{5} + 23 \, A a^{4} b\right )} x\right )} e^{2} + 2 \, {\left (7 \, B b^{5} d x^{5} - 2 \, {\left (32 \, B a b^{4} - 11 \, A b^{5}\right )} d x^{4} - {\left (227 \, B a^{2} b^{3} + 11 \, A a b^{4}\right )} d x^{3} - {\left (227 \, B a^{3} b^{2} + 165 \, A a^{2} b^{3}\right )} d x^{2} - {\left (64 \, B a^{4} b + 209 \, A a^{3} b^{2}\right )} d x + 7 \, {\left (B a^{5} - 11 \, A a^{4} b\right )} d\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{693 \, {\left (b^{3} d^{9} - a^{3} x^{6} e^{9} + 3 \, {\left (a^{2} b d x^{6} - 2 \, a^{3} d x^{5}\right )} e^{8} - 3 \, {\left (a b^{2} d^{2} x^{6} - 6 \, a^{2} b d^{2} x^{5} + 5 \, a^{3} d^{2} x^{4}\right )} e^{7} + {\left (b^{3} d^{3} x^{6} - 18 \, a b^{2} d^{3} x^{5} + 45 \, a^{2} b d^{3} x^{4} - 20 \, a^{3} d^{3} x^{3}\right )} e^{6} + 3 \, {\left (2 \, b^{3} d^{4} x^{5} - 15 \, a b^{2} d^{4} x^{4} + 20 \, a^{2} b d^{4} x^{3} - 5 \, a^{3} d^{4} x^{2}\right )} e^{5} + 3 \, {\left (5 \, b^{3} d^{5} x^{4} - 20 \, a b^{2} d^{5} x^{3} + 15 \, a^{2} b d^{5} x^{2} - 2 \, a^{3} d^{5} x\right )} e^{4} + {\left (20 \, b^{3} d^{6} x^{3} - 45 \, a b^{2} d^{6} x^{2} + 18 \, a^{2} b d^{6} x - a^{3} d^{6}\right )} e^{3} + 3 \, {\left (5 \, b^{3} d^{7} x^{2} - 6 \, a b^{2} d^{7} x + a^{2} b d^{7}\right )} e^{2} + 3 \, {\left (2 \, b^{3} d^{8} x - a b^{2} d^{8}\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. 621 vs.
\(2 (137) = 274\).
time = 1.22, size = 621, normalized size = 4.22 \begin {gather*} \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (7 \, B b^{14} d^{3} {\left | b \right |} e^{6} - 25 \, B a b^{13} d^{2} {\left | b \right |} e^{7} + 4 \, A b^{14} d^{2} {\left | b \right |} e^{7} + 29 \, B a^{2} b^{12} d {\left | b \right |} e^{8} - 8 \, A a b^{13} d {\left | b \right |} e^{8} - 11 \, B a^{3} b^{11} {\left | b \right |} e^{9} + 4 \, A a^{2} b^{12} {\left | b \right |} e^{9}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}} + \frac {11 \, {\left (7 \, B b^{15} d^{4} {\left | b \right |} e^{5} - 32 \, B a b^{14} d^{3} {\left | b \right |} e^{6} + 4 \, A b^{15} d^{3} {\left | b \right |} e^{6} + 54 \, B a^{2} b^{13} d^{2} {\left | b \right |} e^{7} - 12 \, A a b^{14} d^{2} {\left | b \right |} e^{7} - 40 \, B a^{3} b^{12} d {\left | b \right |} e^{8} + 12 \, A a^{2} b^{13} d {\left | b \right |} e^{8} + 11 \, B a^{4} b^{11} {\left | b \right |} e^{9} - 4 \, A a^{3} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} - \frac {99 \, {\left (B a b^{15} d^{4} {\left | b \right |} e^{5} - A b^{16} d^{4} {\left | b \right |} e^{5} - 4 \, B a^{2} b^{14} d^{3} {\left | b \right |} e^{6} + 4 \, A a b^{15} d^{3} {\left | b \right |} e^{6} + 6 \, B a^{3} b^{13} d^{2} {\left | b \right |} e^{7} - 6 \, A a^{2} b^{14} d^{2} {\left | b \right |} e^{7} - 4 \, B a^{4} b^{12} d {\left | b \right |} e^{8} + 4 \, A a^{3} b^{13} d {\left | b \right |} e^{8} + B a^{5} b^{11} {\left | b \right |} e^{9} - A a^{4} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )}^{\frac {7}{2}}}{693 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.73, size = 509, normalized size = 3.46 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (28\,B\,a^5\,d\,e+126\,A\,a^5\,e^2-44\,B\,a^4\,b\,d^2-308\,A\,a^4\,b\,d\,e+198\,A\,a^3\,b^2\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\sqrt {a+b\,x}\,\left (154\,B\,a^5\,e^2-256\,B\,a^4\,b\,d\,e+322\,A\,a^4\,b\,e^2+22\,B\,a^3\,b^2\,d^2-836\,A\,a^3\,b^2\,d\,e+594\,A\,a^2\,b^3\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x^2\,\sqrt {a+b\,x}\,\left (418\,B\,a^4\,b\,e^2-908\,B\,a^3\,b^2\,d\,e+226\,A\,a^3\,b^2\,e^2+330\,B\,a^2\,b^3\,d^2-660\,A\,a^2\,b^3\,d\,e+594\,A\,a\,b^4\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x^3\,\sqrt {a+b\,x}\,\left (330\,B\,a^3\,b^2\,e^2-908\,B\,a^2\,b^3\,d\,e+6\,A\,a^2\,b^3\,e^2+418\,B\,a\,b^4\,d^2-44\,A\,a\,b^4\,d\,e+198\,A\,b^5\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-11\,B\,a\,e+7\,B\,b\,d\right )}{693\,e^5\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,b^3\,x^4\,\left (a\,e-11\,b\,d\right )\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-11\,B\,a\,e+7\,B\,b\,d\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}\right )}{x^6+\frac {d^6}{e^6}+\frac {6\,d\,x^5}{e}+\frac {6\,d^5\,x}{e^5}+\frac {15\,d^2\,x^4}{e^2}+\frac {20\,d^3\,x^3}{e^3}+\frac {15\,d^4\,x^2}{e^4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________