Optimal. Leaf size=255 \[ \frac {32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac {16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac {4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
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Rubi [A] time = 0.16, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ \frac {32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac {16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac {4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {(5 b B d+8 A b e-13 a B e) \int \frac {(a+b x)^{3/2}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {(6 b (5 b B d+8 A b e-13 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {\left (8 b^2 (5 b B d+8 A b e-13 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{429 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {\left (16 b^3 (5 b B d+8 A b e-13 a B e)\right ) \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{3003 e (b d-a e)^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {4 b (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{429 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {32 b^3 (5 b B d+8 A b e-13 a B e) (a+b x)^{5/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 135, normalized size = 0.53 \[ \frac {2 (a+b x)^{5/2} \left (1155 (B d-A e)-\frac {(d+e x) \left (2 b (d+e x) \left (4 b (d+e x) (-5 a e+7 b d+2 b e x)+35 (b d-a e)^2\right )+105 (b d-a e)^3\right ) (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^4}\right )}{15015 e (d+e x)^{13/2} (a e-b d)} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.95, size = 1091, normalized size = 4.28 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 505, normalized size = 1.98 \[ -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (128 A \,b^{4} e^{4} x^{4}-208 B a \,b^{3} e^{4} x^{4}+80 B \,b^{4} d \,e^{3} x^{4}-320 A a \,b^{3} e^{4} x^{3}+832 A \,b^{4} d \,e^{3} x^{3}+520 B \,a^{2} b^{2} e^{4} x^{3}-1552 B a \,b^{3} d \,e^{3} x^{3}+520 B \,b^{4} d^{2} e^{2} x^{3}+560 A \,a^{2} b^{2} e^{4} x^{2}-2080 A a \,b^{3} d \,e^{3} x^{2}+2288 A \,b^{4} d^{2} e^{2} x^{2}-910 B \,a^{3} b \,e^{4} x^{2}+3730 B \,a^{2} b^{2} d \,e^{3} x^{2}-5018 B a \,b^{3} d^{2} e^{2} x^{2}+1430 B \,b^{4} d^{3} e \,x^{2}-840 A \,a^{3} b \,e^{4} x +3640 A \,a^{2} b^{2} d \,e^{3} x -5720 A a \,b^{3} d^{2} e^{2} x +3432 A \,b^{4} d^{3} e x +1365 B \,a^{4} e^{4} x -6440 B \,a^{3} b d \,e^{3} x +11570 B \,a^{2} b^{2} d^{2} e^{2} x -9152 B a \,b^{3} d^{3} e x +2145 B \,b^{4} d^{4} x +1155 A \,a^{4} e^{4}-5460 A \,a^{3} b d \,e^{3}+10010 A \,a^{2} b^{2} d^{2} e^{2}-8580 A a \,b^{3} d^{3} e +3003 A \,b^{4} d^{4}+210 B \,a^{4} d \,e^{3}-910 B \,a^{3} b \,d^{2} e^{2}+1430 B \,a^{2} b^{2} d^{3} e -858 B a \,b^{3} d^{4}\right )}{15015 \left (e x +d \right )^{\frac {13}{2}} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.11, size = 752, normalized size = 2.95 \[ -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (420\,B\,a^6\,d\,e^3+2310\,A\,a^6\,e^4-1820\,B\,a^5\,b\,d^2\,e^2-10920\,A\,a^5\,b\,d\,e^3+2860\,B\,a^4\,b^2\,d^3\,e+20020\,A\,a^4\,b^2\,d^2\,e^2-1716\,B\,a^3\,b^3\,d^4-17160\,A\,a^3\,b^3\,d^3\,e+6006\,A\,a^2\,b^4\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {x\,\sqrt {a+b\,x}\,\left (2730\,B\,a^6\,e^4-12040\,B\,a^5\,b\,d\,e^3+2940\,A\,a^5\,b\,e^4+19500\,B\,a^4\,b^2\,d^2\,e^2-14560\,A\,a^4\,b^2\,d\,e^3-12584\,B\,a^3\,b^3\,d^3\,e+28600\,A\,a^3\,b^3\,d^2\,e^2+858\,B\,a^2\,b^4\,d^4-27456\,A\,a^2\,b^4\,d^3\,e+12012\,A\,a\,b^5\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {x^2\,\sqrt {a+b\,x}\,\left (3640\,B\,a^5\,b\,e^4-17880\,B\,a^4\,b^2\,d\,e^3+70\,A\,a^4\,b^2\,e^4+34424\,B\,a^3\,b^3\,d^2\,e^2-520\,A\,a^3\,b^3\,d\,e^3-30888\,B\,a^2\,b^4\,d^3\,e+1716\,A\,a^2\,b^4\,d^2\,e^2+6864\,B\,a\,b^5\,d^4-3432\,A\,a\,b^5\,d^3\,e+6006\,A\,b^6\,d^4\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^4\,{\left (a\,e-b\,d\right )}^5}-\frac {2\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )\,\left (5\,a^3\,e^3-39\,a^2\,b\,d\,e^2+143\,a\,b^2\,d^2\,e-429\,b^3\,d^3\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}-\frac {16\,b^4\,x^5\,\left (a\,e-13\,b\,d\right )\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^5\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-26\,a\,b\,d\,e+143\,b^2\,d^2\right )\,\left (8\,A\,b\,e-13\,B\,a\,e+5\,B\,b\,d\right )}{15015\,e^6\,{\left (a\,e-b\,d\right )}^5}\right )}{x^7+\frac {d^7}{e^7}+\frac {7\,d\,x^6}{e}+\frac {7\,d^6\,x}{e^6}+\frac {21\,d^2\,x^5}{e^2}+\frac {35\,d^3\,x^4}{e^3}+\frac {35\,d^4\,x^3}{e^4}+\frac {21\,d^5\,x^2}{e^5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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