3.5.45 \(\int \sqrt {x} (a+b x)^3 \, dx\) [445]
Optimal. Leaf size=51 \[ \frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \]
[Out]
2/3*a^3*x^(3/2)+6/5*a^2*b*x^(5/2)+6/7*a*b^2*x^(7/2)+2/9*b^3*x^(9/2)
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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} \frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[x]*(a + b*x)^3,x]
[Out]
(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(5/2))/5 + (6*a*b^2*x^(7/2))/7 + (2*b^3*x^(9/2))/9
Rule 45
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin {align*} \int \sqrt {x} (a+b x)^3 \, dx &=\int \left (a^3 \sqrt {x}+3 a^2 b x^{3/2}+3 a b^2 x^{5/2}+b^3 x^{7/2}\right ) \, dx\\ &=\frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.76 \begin {gather*} \frac {2}{315} x^{3/2} \left (105 a^3+189 a^2 b x+135 a b^2 x^2+35 b^3 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[x]*(a + b*x)^3,x]
[Out]
(2*x^(3/2)*(105*a^3 + 189*a^2*b*x + 135*a*b^2*x^2 + 35*b^3*x^3))/315
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 27.15, size = 2556, normalized size = 50.12
result too large to display
Warning: Unable to verify antiderivative.
[In]
mathics('Integrate[Sqrt[x]*(a + b*x)^3,x]')
[Out]
Piecewise[{{2 Sqrt[a] (16 I a ^ 4 + 105 a ^ 3 b x Sqrt[b x / a] + 189 a ^ 2 b ^ 2 x ^ 2 Sqrt[b x / a] + 135 a
b ^ 3 x ^ 3 Sqrt[b x / a] + 35 b ^ 4 x ^ 4 Sqrt[b x / a]) / (315 b ^ (3 / 2)), Abs[(a + b x) / a] > 1}}, -32 I
a ^ (49 / 2) Sqrt[1 - b (a / b + x) / a] / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 47
25 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a
/ b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) + I 32 a ^
(49 / 2) / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x)
^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (1
3 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) - 192 I a ^ (47 / 2) b (a / b + x) / (315 a
^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17
b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x)
^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) + I 176 a ^ (47 / 2) b (a / b + x) Sqrt[1 - b (a / b + x) / a]
/ (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 6
300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (
a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) - 396 I a ^ (45 / 2) b ^ 2 Sqrt[1 - b (a / b + x) /
a] (a / b + x) ^ 2 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (
a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^
15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) + I 480 a ^ (45 / 2) b ^ 2 (a / b
+ x) ^ 2 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x)
^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (1
3 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) - 640 I a ^ (43 / 2) b ^ 3 (a / b + x) ^ 3 /
(315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 630
0 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a
/ b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) + I 462 a ^ (43 / 2) b ^ 3 Sqrt[1 - b (a / b + x) / a]
(a / b + x) ^ 3 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a
/ b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 1
5 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) - 210 I a ^ (41 / 2) b ^ 4 Sqrt[1 -
b (a / b + x) / a] (a / b + x) ^ 4 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^
18 b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x
) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) + I 480 a ^ (41 /
2) b ^ 4 (a / b + x) ^ 4 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7
/ 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 18
90 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) - 378 I a ^ (39 / 2) b ^ 5 S
qrt[1 - b (a / b + x) / a] (a / b + x) ^ 5 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4
725 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a
/ b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) - 192 I a
^ (39 / 2) b ^ 5 (a / b + x) ^ 5 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18
b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x)
^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) + I 32 a ^ (37 / 2)
b ^ 6 (a / b + x) ^ 6 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2
) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890
a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) + I 1134 a ^ (37 / 2) b ^ 6 Sqr
t[1 - b (a / b + x) / a] (a / b + x) ^ 6 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 472
5 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a /
b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) - 1494 I a ^
(35 / 2) b ^ 7 Sqrt[1 - b (a / b + x) / a] (a / b + x) ^ 7 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2
) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 1
6 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x
) ^ 6) + I 1098 a ^ (33 / 2) b ^ 8 Sqrt[1 - b (a / b + x) / a] (a / b + x) ^ 8 / (315 a ^ 20 b ^ (3 / 2) - 189
0 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^ (9 / 2) (a / b +
x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5 + 315 a ^ 14 b ^
(15 / 2) (a / b + x) ^ 6) - 430 I a ^ (31 / 2) b ^ 9 Sqrt[1 - b (a / b + x) / a] (a / b + x) ^ 9 / (315 a ^ 20
b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x) ^ 2 - 6300 a ^ 17 b ^
(9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (13 / 2) (a / b + x) ^ 5
+ 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6) + I 70 a ^ (29 / 2) b ^ 10 Sqrt[1 - b (a / b + x) / a] (a / b + x)
^ 10 / (315 a ^ 20 b ^ (3 / 2) - 1890 a ^ 19 b ^ (5 / 2) (a / b + x) + 4725 a ^ 18 b ^ (7 / 2) (a / b + x) ^
2 - 6300 a ^ 17 b ^ (9 / 2) (a / b + x) ^ 3 + 4725 a ^ 16 b ^ (11 / 2) (a / b + x) ^ 4 - 1890 a ^ 15 b ^ (13 /
2) (a / b + x) ^ 5 + 315 a ^ 14 b ^ (15 / 2) (a / b + x) ^ 6)]
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Maple [A]
time = 0.09, size = 36, normalized size = 0.71
| | |
| method |
result |
size |
| | |
| gosper |
\(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) |
\(36\) |
| derivativedivides |
\(\frac {2 a^{3} x^{\frac {3}{2}}}{3}+\frac {6 a^{2} b \,x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\) |
\(36\) |
| default |
\(\frac {2 a^{3} x^{\frac {3}{2}}}{3}+\frac {6 a^{2} b \,x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\) |
\(36\) |
| trager |
\(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) |
\(36\) |
| risch |
\(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) |
\(36\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3*x^(1/2),x,method=_RETURNVERBOSE)
[Out]
2/3*a^3*x^(3/2)+6/5*a^2*b*x^(5/2)+6/7*a*b^2*x^(7/2)+2/9*b^3*x^(9/2)
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Maxima [A]
time = 0.25, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*x^(1/2),x, algorithm="maxima")
[Out]
2/9*b^3*x^(9/2) + 6/7*a*b^2*x^(7/2) + 6/5*a^2*b*x^(5/2) + 2/3*a^3*x^(3/2)
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Fricas [A]
time = 0.30, size = 38, normalized size = 0.75 \begin {gather*} \frac {2}{315} \, {\left (35 \, b^{3} x^{4} + 135 \, a b^{2} x^{3} + 189 \, a^{2} b x^{2} + 105 \, a^{3} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*x^(1/2),x, algorithm="fricas")
[Out]
2/315*(35*b^3*x^4 + 135*a*b^2*x^3 + 189*a^2*b*x^2 + 105*a^3*x)*sqrt(x)
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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]
integrate((b*x+a)**3*x**(1/2),x)
[Out]
Timed out
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Giac [A]
time = 0.00, size = 61, normalized size = 1.20 \begin {gather*} \frac {2}{9} b^{3} \sqrt {x} x^{4}+\frac {6}{7} a b^{2} \sqrt {x} x^{3}+\frac {6}{5} a^{2} b \sqrt {x} x^{2}+\frac {2}{3} a^{3} \sqrt {x} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*x^(1/2),x)
[Out]
2/9*b^3*x^(9/2) + 6/7*a*b^2*x^(7/2) + 6/5*a^2*b*x^(5/2) + 2/3*a^3*x^(3/2)
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Mupad [B]
time = 0.04, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,a^3\,x^{3/2}}{3}+\frac {2\,b^3\,x^{9/2}}{9}+\frac {6\,a^2\,b\,x^{5/2}}{5}+\frac {6\,a\,b^2\,x^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(1/2)*(a + b*x)^3,x)
[Out]
(2*a^3*x^(3/2))/3 + (2*b^3*x^(9/2))/9 + (6*a^2*b*x^(5/2))/5 + (6*a*b^2*x^(7/2))/7
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