Optimal. Leaf size=129 \[ -\frac{8 b^3 (c+d x)^{11/2} (b c-a d)}{11 d^5}+\frac{4 b^2 (c+d x)^{9/2} (b c-a d)^2}{3 d^5}-\frac{8 b (c+d x)^{7/2} (b c-a d)^3}{7 d^5}+\frac{2 (c+d x)^{5/2} (b c-a d)^4}{5 d^5}+\frac{2 b^4 (c+d x)^{13/2}}{13 d^5} \]
[Out]
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Rubi [A] time = 0.118342, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{8 b^3 (c+d x)^{11/2} (b c-a d)}{11 d^5}+\frac{4 b^2 (c+d x)^{9/2} (b c-a d)^2}{3 d^5}-\frac{8 b (c+d x)^{7/2} (b c-a d)^3}{7 d^5}+\frac{2 (c+d x)^{5/2} (b c-a d)^4}{5 d^5}+\frac{2 b^4 (c+d x)^{13/2}}{13 d^5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^4*(c + d*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 31.3134, size = 119, normalized size = 0.92 \[ \frac{2 b^{4} \left (c + d x\right )^{\frac{13}{2}}}{13 d^{5}} + \frac{8 b^{3} \left (c + d x\right )^{\frac{11}{2}} \left (a d - b c\right )}{11 d^{5}} + \frac{4 b^{2} \left (c + d x\right )^{\frac{9}{2}} \left (a d - b c\right )^{2}}{3 d^{5}} + \frac{8 b \left (c + d x\right )^{\frac{7}{2}} \left (a d - b c\right )^{3}}{7 d^{5}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )^{4}}{5 d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**4*(d*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.145301, size = 154, normalized size = 1.19 \[ \frac{2 (c+d x)^{5/2} \left (3003 a^4 d^4+1716 a^3 b d^3 (5 d x-2 c)+286 a^2 b^2 d^2 \left (8 c^2-20 c d x+35 d^2 x^2\right )+52 a b^3 d \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )+b^4 \left (128 c^4-320 c^3 d x+560 c^2 d^2 x^2-840 c d^3 x^3+1155 d^4 x^4\right )\right )}{15015 d^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^4*(c + d*x)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 186, normalized size = 1.4 \[{\frac{2310\,{x}^{4}{b}^{4}{d}^{4}+10920\,a{b}^{3}{d}^{4}{x}^{3}-1680\,{b}^{4}c{d}^{3}{x}^{3}+20020\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-7280\,a{b}^{3}c{d}^{3}{x}^{2}+1120\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+17160\,{a}^{3}b{d}^{4}x-11440\,{a}^{2}{b}^{2}c{d}^{3}x+4160\,a{b}^{3}{c}^{2}{d}^{2}x-640\,{b}^{4}{c}^{3}dx+6006\,{a}^{4}{d}^{4}-6864\,{a}^{3}bc{d}^{3}+4576\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-1664\,a{b}^{3}{c}^{3}d+256\,{b}^{4}{c}^{4}}{15015\,{d}^{5}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^4*(d*x+c)^(3/2),x)
[Out]
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Maxima [A] time = 1.34686, size = 244, normalized size = 1.89 \[ \frac{2 \,{\left (1155 \,{\left (d x + c\right )}^{\frac{13}{2}} b^{4} - 5460 \,{\left (b^{4} c - a b^{3} d\right )}{\left (d x + c\right )}^{\frac{11}{2}} + 10010 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{9}{2}} - 8580 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 3003 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left (d x + c\right )}^{\frac{5}{2}}\right )}}{15015 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4*(d*x + c)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206677, size = 420, normalized size = 3.26 \[ \frac{2 \,{\left (1155 \, b^{4} d^{6} x^{6} + 128 \, b^{4} c^{6} - 832 \, a b^{3} c^{5} d + 2288 \, a^{2} b^{2} c^{4} d^{2} - 3432 \, a^{3} b c^{3} d^{3} + 3003 \, a^{4} c^{2} d^{4} + 210 \,{\left (7 \, b^{4} c d^{5} + 26 \, a b^{3} d^{6}\right )} x^{5} + 35 \,{\left (b^{4} c^{2} d^{4} + 208 \, a b^{3} c d^{5} + 286 \, a^{2} b^{2} d^{6}\right )} x^{4} - 20 \,{\left (2 \, b^{4} c^{3} d^{3} - 13 \, a b^{3} c^{2} d^{4} - 715 \, a^{2} b^{2} c d^{5} - 429 \, a^{3} b d^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{4} c^{4} d^{2} - 104 \, a b^{3} c^{3} d^{3} + 286 \, a^{2} b^{2} c^{2} d^{4} + 4576 \, a^{3} b c d^{5} + 1001 \, a^{4} d^{6}\right )} x^{2} - 2 \,{\left (32 \, b^{4} c^{5} d - 208 \, a b^{3} c^{4} d^{2} + 572 \, a^{2} b^{2} c^{3} d^{3} - 858 \, a^{3} b c^{2} d^{4} - 3003 \, a^{4} c d^{5}\right )} x\right )} \sqrt{d x + c}}{15015 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4*(d*x + c)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.97796, size = 559, normalized size = 4.33 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**4*(d*x+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227806, size = 761, normalized size = 5.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4*(d*x + c)^(3/2),x, algorithm="giac")
[Out]