Optimal. Leaf size=204 \[ -\frac {2 a^7 \left (a+b \sqrt {x}\right )^{1+p}}{b^8 (1+p)}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{2+p}}{b^8 (2+p)}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{3+p}}{b^8 (3+p)}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{4+p}}{b^8 (4+p)}-\frac {70 a^3 \left (a+b \sqrt {x}\right )^{5+p}}{b^8 (5+p)}+\frac {42 a^2 \left (a+b \sqrt {x}\right )^{6+p}}{b^8 (6+p)}-\frac {14 a \left (a+b \sqrt {x}\right )^{7+p}}{b^8 (7+p)}+\frac {2 \left (a+b \sqrt {x}\right )^{8+p}}{b^8 (8+p)} \]
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Rubi [A]
time = 0.08, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} -\frac {2 a^7 \left (a+b \sqrt {x}\right )^{p+1}}{b^8 (p+1)}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{p+2}}{b^8 (p+2)}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{p+3}}{b^8 (p+3)}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{p+4}}{b^8 (p+4)}-\frac {70 a^3 \left (a+b \sqrt {x}\right )^{p+5}}{b^8 (p+5)}+\frac {42 a^2 \left (a+b \sqrt {x}\right )^{p+6}}{b^8 (p+6)}-\frac {14 a \left (a+b \sqrt {x}\right )^{p+7}}{b^8 (p+7)}+\frac {2 \left (a+b \sqrt {x}\right )^{p+8}}{b^8 (p+8)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^p x^3 \, dx &=2 \text {Subst}\left (\int x^7 (a+b x)^p \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (-\frac {a^7 (a+b x)^p}{b^7}+\frac {7 a^6 (a+b x)^{1+p}}{b^7}-\frac {21 a^5 (a+b x)^{2+p}}{b^7}+\frac {35 a^4 (a+b x)^{3+p}}{b^7}-\frac {35 a^3 (a+b x)^{4+p}}{b^7}+\frac {21 a^2 (a+b x)^{5+p}}{b^7}-\frac {7 a (a+b x)^{6+p}}{b^7}+\frac {(a+b x)^{7+p}}{b^7}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 a^7 \left (a+b \sqrt {x}\right )^{1+p}}{b^8 (1+p)}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{2+p}}{b^8 (2+p)}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{3+p}}{b^8 (3+p)}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{4+p}}{b^8 (4+p)}-\frac {70 a^3 \left (a+b \sqrt {x}\right )^{5+p}}{b^8 (5+p)}+\frac {42 a^2 \left (a+b \sqrt {x}\right )^{6+p}}{b^8 (6+p)}-\frac {14 a \left (a+b \sqrt {x}\right )^{7+p}}{b^8 (7+p)}+\frac {2 \left (a+b \sqrt {x}\right )^{8+p}}{b^8 (8+p)}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 168, normalized size = 0.82 \begin {gather*} \frac {2 \left (-\frac {a^7}{1+p}+\frac {7 a^6 \left (a+b \sqrt {x}\right )}{2+p}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^2}{3+p}+\frac {35 a^4 \left (a+b \sqrt {x}\right )^3}{4+p}-\frac {35 a^3 \left (a+b \sqrt {x}\right )^4}{5+p}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^5}{6+p}-\frac {7 a \left (a+b \sqrt {x}\right )^6}{7+p}+\frac {\left (a+b \sqrt {x}\right )^7}{8+p}\right ) \left (a+b \sqrt {x}\right )^{1+p}}{b^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \sqrt {x}\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 286, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left ({\left (p^{7} + 28 \, p^{6} + 322 \, p^{5} + 1960 \, p^{4} + 6769 \, p^{3} + 13132 \, p^{2} + 13068 \, p + 5040\right )} b^{8} x^{4} + {\left (p^{7} + 21 \, p^{6} + 175 \, p^{5} + 735 \, p^{4} + 1624 \, p^{3} + 1764 \, p^{2} + 720 \, p\right )} a b^{7} x^{\frac {7}{2}} - 7 \, {\left (p^{6} + 15 \, p^{5} + 85 \, p^{4} + 225 \, p^{3} + 274 \, p^{2} + 120 \, p\right )} a^{2} b^{6} x^{3} + 42 \, {\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )} a^{3} b^{5} x^{\frac {5}{2}} - 210 \, {\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )} a^{4} b^{4} x^{2} + 840 \, {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a^{5} b^{3} x^{\frac {3}{2}} - 2520 \, {\left (p^{2} + p\right )} a^{6} b^{2} x + 5040 \, a^{7} b p \sqrt {x} - 5040 \, a^{8}\right )} {\left (b \sqrt {x} + a\right )}^{p}}{{\left (p^{8} + 36 \, p^{7} + 546 \, p^{6} + 4536 \, p^{5} + 22449 \, p^{4} + 67284 \, p^{3} + 118124 \, p^{2} + 109584 \, p + 40320\right )} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 458 vs.
\(2 (188) = 376\).
time = 0.43, size = 458, normalized size = 2.25 \begin {gather*} -\frac {2 \, {\left (5040 \, a^{8} - {\left (b^{8} p^{7} + 28 \, b^{8} p^{6} + 322 \, b^{8} p^{5} + 1960 \, b^{8} p^{4} + 6769 \, b^{8} p^{3} + 13132 \, b^{8} p^{2} + 13068 \, b^{8} p + 5040 \, b^{8}\right )} x^{4} + 7 \, {\left (a^{2} b^{6} p^{6} + 15 \, a^{2} b^{6} p^{5} + 85 \, a^{2} b^{6} p^{4} + 225 \, a^{2} b^{6} p^{3} + 274 \, a^{2} b^{6} p^{2} + 120 \, a^{2} b^{6} p\right )} x^{3} + 210 \, {\left (a^{4} b^{4} p^{4} + 6 \, a^{4} b^{4} p^{3} + 11 \, a^{4} b^{4} p^{2} + 6 \, a^{4} b^{4} p\right )} x^{2} + 2520 \, {\left (a^{6} b^{2} p^{2} + a^{6} b^{2} p\right )} x - {\left (5040 \, a^{7} b p + {\left (a b^{7} p^{7} + 21 \, a b^{7} p^{6} + 175 \, a b^{7} p^{5} + 735 \, a b^{7} p^{4} + 1624 \, a b^{7} p^{3} + 1764 \, a b^{7} p^{2} + 720 \, a b^{7} p\right )} x^{3} + 42 \, {\left (a^{3} b^{5} p^{5} + 10 \, a^{3} b^{5} p^{4} + 35 \, a^{3} b^{5} p^{3} + 50 \, a^{3} b^{5} p^{2} + 24 \, a^{3} b^{5} p\right )} x^{2} + 840 \, {\left (a^{5} b^{3} p^{3} + 3 \, a^{5} b^{3} p^{2} + 2 \, a^{5} b^{3} p\right )} x\right )} \sqrt {x}\right )} {\left (b \sqrt {x} + a\right )}^{p}}{b^{8} p^{8} + 36 \, b^{8} p^{7} + 546 \, b^{8} p^{6} + 4536 \, b^{8} p^{5} + 22449 \, b^{8} p^{4} + 67284 \, b^{8} p^{3} + 118124 \, b^{8} p^{2} + 109584 \, b^{8} p + 40320 \, b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1642 vs.
\(2 (188) = 376\).
time = 1.29, size = 1642, normalized size = 8.05 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.11, size = 590, normalized size = 2.89 \begin {gather*} {\left (a+b\,\sqrt {x}\right )}^p\,\left (\frac {2\,x^4\,\left (p^7+28\,p^6+322\,p^5+1960\,p^4+6769\,p^3+13132\,p^2+13068\,p+5040\right )}{p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320}-\frac {10080\,a^8}{b^8\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {10080\,a^7\,p\,\sqrt {x}}{b^7\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {5040\,a^6\,p\,x\,\left (p+1\right )}{b^6\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {14\,a^2\,p\,x^3\,\left (p^5+15\,p^4+85\,p^3+225\,p^2+274\,p+120\right )}{b^2\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {1680\,a^5\,p\,x^{3/2}\,\left (p^2+3\,p+2\right )}{b^5\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}-\frac {420\,a^4\,p\,x^2\,\left (p^3+6\,p^2+11\,p+6\right )}{b^4\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {2\,a\,p\,x^{7/2}\,\left (p^6+21\,p^5+175\,p^4+735\,p^3+1624\,p^2+1764\,p+720\right )}{b\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}+\frac {84\,a^3\,p\,x^{5/2}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}{b^3\,\left (p^8+36\,p^7+546\,p^6+4536\,p^5+22449\,p^4+67284\,p^3+118124\,p^2+109584\,p+40320\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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