Optimal. Leaf size=309 \[ \frac {256 b^4 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{45045 e (d+e x)^{3/2} (b d-a e)^6}+\frac {128 b^3 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac {32 b^2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac {16 b (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \begin {gather*} \frac {256 b^4 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{45045 e (d+e x)^{3/2} (b d-a e)^6}+\frac {128 b^3 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac {32 b^2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac {16 b (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {(3 b B d+10 A b e-13 a B e) \int \frac {\sqrt {a+b x}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {(8 b (3 b B d+10 A b e-13 a B e)) \int \frac {\sqrt {a+b x}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {\left (16 b^2 (3 b B d+10 A b e-13 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{9/2}} \, dx}{429 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {\left (64 b^3 (3 b B d+10 A b e-13 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}} \, dx}{3003 e (b d-a e)^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {128 b^3 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}+\frac {\left (128 b^4 (3 b B d+10 A b e-13 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}} \, dx}{15015 e (b d-a e)^5}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {128 b^3 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}+\frac {256 b^4 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{45045 e (b d-a e)^6 (d+e x)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.36, size = 160, normalized size = 0.52 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (3465 (B d-A e)-\frac {2 (d+e x) \left (8 b (d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-3 a e+5 b d+2 b e x)+15 (b d-a e)^2\right )+35 (b d-a e)^3\right )+315 (b d-a e)^4\right ) \left (-\frac {13 a B e}{2}+5 A b e+\frac {3 b B d}{2}\right )}{(b d-a e)^5}\right )}{45045 e (d+e x)^{13/2} (a e-b d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.32, size = 347, normalized size = 1.12 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (-\frac {45045 A b^4 e (a+b x)}{d+e x}+\frac {64350 A b^3 e^2 (a+b x)^2}{(d+e x)^2}-\frac {50050 A b^2 e^3 (a+b x)^3}{(d+e x)^3}-\frac {3465 A e^5 (a+b x)^5}{(d+e x)^5}+\frac {20475 A b e^4 (a+b x)^4}{(d+e x)^4}+\frac {9009 b^4 B d (a+b x)}{d+e x}-15015 a b^4 B+\frac {36036 a b^3 B e (a+b x)}{d+e x}-\frac {25740 b^3 B d e (a+b x)^2}{(d+e x)^2}-\frac {38610 a b^2 B e^2 (a+b x)^2}{(d+e x)^2}+\frac {30030 b^2 B d e^2 (a+b x)^3}{(d+e x)^3}-\frac {4095 a B e^4 (a+b x)^4}{(d+e x)^4}+\frac {3465 B d e^4 (a+b x)^5}{(d+e x)^5}+\frac {20020 a b B e^3 (a+b x)^3}{(d+e x)^3}-\frac {16380 b B d e^3 (a+b x)^4}{(d+e x)^4}+15015 A b^5\right )}{45045 (d+e x)^{3/2} (b d-a e)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] 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] time = 5.76, size = 1216, normalized size = 3.94
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 722, normalized size = 2.34 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} e^{5} x^{5}+1664 B a \,b^{4} e^{5} x^{5}-384 B \,b^{5} d \,e^{4} x^{5}+1920 A a \,b^{4} e^{5} x^{4}-8320 A \,b^{5} d \,e^{4} x^{4}-2496 B \,a^{2} b^{3} e^{5} x^{4}+11392 B a \,b^{4} d \,e^{4} x^{4}-2496 B \,b^{5} d^{2} e^{3} x^{4}-2400 A \,a^{2} b^{3} e^{5} x^{3}+12480 A a \,b^{4} d \,e^{4} x^{3}-22880 A \,b^{5} d^{2} e^{3} x^{3}+3120 B \,a^{3} b^{2} e^{5} x^{3}-16944 B \,a^{2} b^{3} d \,e^{4} x^{3}+33488 B a \,b^{4} d^{2} e^{3} x^{3}-6864 B \,b^{5} d^{3} e^{2} x^{3}+2800 A \,a^{3} b^{2} e^{5} x^{2}-15600 A \,a^{2} b^{3} d \,e^{4} x^{2}+34320 A a \,b^{4} d^{2} e^{3} x^{2}-34320 A \,b^{5} d^{3} e^{2} x^{2}-3640 B \,a^{4} b \,e^{5} x^{2}+21120 B \,a^{3} b^{2} d \,e^{4} x^{2}-49296 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+54912 B a \,b^{4} d^{3} e^{2} x^{2}-10296 B \,b^{5} d^{4} e \,x^{2}-3150 A \,a^{4} b \,e^{5} x +18200 A \,a^{3} b^{2} d \,e^{4} x -42900 A \,a^{2} b^{3} d^{2} e^{3} x +51480 A a \,b^{4} d^{3} e^{2} x -30030 A \,b^{5} d^{4} e x +4095 B \,a^{5} e^{5} x -24605 B \,a^{4} b d \,e^{4} x +61230 B \,a^{3} b^{2} d^{2} e^{3} x -79794 B \,a^{2} b^{3} d^{3} e^{2} x +54483 B a \,b^{4} d^{4} e x -9009 B \,b^{5} d^{5} x +3465 A \,a^{5} e^{5}-20475 A \,a^{4} b d \,e^{4}+50050 A \,a^{3} b^{2} d^{2} e^{3}-64350 A \,a^{2} b^{3} d^{3} e^{2}+45045 A a \,b^{4} d^{4} e -15015 A \,b^{5} d^{5}+630 B \,a^{5} d \,e^{4}-3640 B \,a^{4} b \,d^{2} e^{3}+8580 B \,a^{3} b^{2} d^{3} e^{2}-10296 B \,a^{2} b^{3} d^{4} e +6006 B a \,b^{4} d^{5}\right )}{45045 \left (e x +d \right )^{\frac {13}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )} \end {gather*}
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]
________________________________________________________________________________________
mupad [B] time = 3.03, size = 750, normalized size = 2.43 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {x\,\sqrt {a+b\,x}\,\left (-8190\,B\,a^6\,e^5+47950\,B\,a^5\,b\,d\,e^4-630\,A\,a^5\,b\,e^5-115180\,B\,a^4\,b^2\,d^2\,e^3+4550\,A\,a^4\,b^2\,d\,e^4+142428\,B\,a^3\,b^3\,d^3\,e^2-14300\,A\,a^3\,b^3\,d^2\,e^3-88374\,B\,a^2\,b^4\,d^4\,e+25740\,A\,a^2\,b^4\,d^3\,e^2+6006\,B\,a\,b^5\,d^5-30030\,A\,a\,b^5\,d^4\,e+30030\,A\,b^6\,d^5\right )}{45045\,e^7\,{\left (a\,e-b\,d\right )}^6}-\frac {\sqrt {a+b\,x}\,\left (1260\,B\,a^6\,d\,e^4+6930\,A\,a^6\,e^5-7280\,B\,a^5\,b\,d^2\,e^3-40950\,A\,a^5\,b\,d\,e^4+17160\,B\,a^4\,b^2\,d^3\,e^2+100100\,A\,a^4\,b^2\,d^2\,e^3-20592\,B\,a^3\,b^3\,d^4\,e-128700\,A\,a^3\,b^3\,d^3\,e^2+12012\,B\,a^2\,b^4\,d^5+90090\,A\,a^2\,b^4\,d^4\,e-30030\,A\,a\,b^5\,d^5\right )}{45045\,e^7\,{\left (a\,e-b\,d\right )}^6}+\frac {256\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )}{45045\,e^3\,{\left (a\,e-b\,d\right )}^6}-\frac {16\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )\,\left (5\,a^3\,e^3-39\,a^2\,b\,d\,e^2+143\,a\,b^2\,d^2\,e-429\,b^3\,d^3\right )}{45045\,e^6\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,x^2\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )\,\left (35\,a^4\,e^4-260\,a^3\,b\,d\,e^3+858\,a^2\,b^2\,d^2\,e^2-1716\,a\,b^3\,d^3\,e+3003\,b^4\,d^4\right )}{45045\,e^7\,{\left (a\,e-b\,d\right )}^6}-\frac {128\,b^4\,x^5\,\left (a\,e-13\,b\,d\right )\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )}{45045\,e^4\,{\left (a\,e-b\,d\right )}^6}+\frac {32\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-26\,a\,b\,d\,e+143\,b^2\,d^2\right )\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )}{45045\,e^5\,{\left (a\,e-b\,d\right )}^6}\right )}{x^7+\frac {d^7}{e^7}+\frac {7\,d\,x^6}{e}+\frac {7\,d^6\,x}{e^6}+\frac {21\,d^2\,x^5}{e^2}+\frac {35\,d^3\,x^4}{e^3}+\frac {35\,d^4\,x^3}{e^4}+\frac {21\,d^5\,x^2}{e^5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] 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]
________________________________________________________________________________________