Optimal. Leaf size=173 \[ -\frac{2 b^2 (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5}+\frac{2 b (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5}-\frac{2 (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5}+\frac{2 b^3 B (d+e x)^{13/2}}{13 e^5} \]
[Out]
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Rubi [A] time = 0.215212, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 b^2 (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5}+\frac{2 b (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5}-\frac{2 (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5}+\frac{2 b^3 B (d+e x)^{13/2}}{13 e^5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^3*(A + B*x)*(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 47.5374, size = 170, normalized size = 0.98 \[ \frac{2 B b^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{2 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (A b e + 3 B a e - 4 B b d\right )}{11 e^{5}} + \frac{2 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{3 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{7 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{3}}{5 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.354477, size = 228, normalized size = 1.32 \[ \frac{2 (d+e x)^{5/2} \left (429 a^3 e^3 (7 A e-2 B d+5 B e x)+143 a^2 b e^2 \left (9 A e (5 e x-2 d)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-13 a b^2 e \left (3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )-11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+b^3 \left (13 A e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+B \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )\right )}{15015 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 301, normalized size = 1.7 \[{\frac{2310\,B{b}^{3}{x}^{4}{e}^{4}+2730\,A{b}^{3}{e}^{4}{x}^{3}+8190\,Ba{b}^{2}{e}^{4}{x}^{3}-1680\,B{b}^{3}d{e}^{3}{x}^{3}+10010\,Aa{b}^{2}{e}^{4}{x}^{2}-1820\,A{b}^{3}d{e}^{3}{x}^{2}+10010\,B{a}^{2}b{e}^{4}{x}^{2}-5460\,Ba{b}^{2}d{e}^{3}{x}^{2}+1120\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+12870\,A{a}^{2}b{e}^{4}x-5720\,Aa{b}^{2}d{e}^{3}x+1040\,A{b}^{3}{d}^{2}{e}^{2}x+4290\,B{a}^{3}{e}^{4}x-5720\,B{a}^{2}bd{e}^{3}x+3120\,Ba{b}^{2}{d}^{2}{e}^{2}x-640\,B{b}^{3}{d}^{3}ex+6006\,{a}^{3}A{e}^{4}-5148\,A{a}^{2}bd{e}^{3}+2288\,Aa{b}^{2}{d}^{2}{e}^{2}-416\,A{b}^{3}{d}^{3}e-1716\,B{a}^{3}d{e}^{3}+2288\,B{a}^{2}b{d}^{2}{e}^{2}-1248\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{15015\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3*(B*x+A)*(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 1.34576, size = 358, normalized size = 2.07 \[ \frac{2 \,{\left (1155 \,{\left (e x + d\right )}^{\frac{13}{2}} B b^{3} - 1365 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 5005 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 2145 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 3003 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{15015 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230838, size = 602, normalized size = 3.48 \[ \frac{2 \,{\left (1155 \, B b^{3} e^{6} x^{6} + 128 \, B b^{3} d^{6} + 3003 \, A a^{3} d^{2} e^{4} - 208 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e + 1144 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{2} - 858 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{3} + 105 \,{\left (14 \, B b^{3} d e^{5} + 13 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{6}\right )} x^{5} + 35 \,{\left (B b^{3} d^{2} e^{4} + 52 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{5} + 143 \,{\left (B a^{2} b + A a b^{2}\right )} e^{6}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{3} e^{3} - 13 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{4} - 1430 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{5} - 429 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{4} e^{2} + 1001 \, A a^{3} e^{6} - 26 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{3} + 143 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{4} + 1144 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{5} e - 6006 \, A a^{3} d e^{5} - 104 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{2} + 572 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{3} - 429 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.0982, size = 913, normalized size = 5.28 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.225424, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(3/2),x, algorithm="giac")
[Out]