Optimal. Leaf size=350 \[ -\frac {2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{1+p}}{b^8 d^4 (1+p)}+\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{2+p}}{b^8 d^4 (2+p)}-\frac {6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3+p}}{b^8 d^4 (3+p)}+\frac {2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{4+p}}{b^8 d^4 (4+p)}-\frac {10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5+p}}{b^8 d^4 (5+p)}+\frac {6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{6+p}}{b^8 d^4 (6+p)}-\frac {14 a \left (a+b \sqrt {c+d x}\right )^{7+p}}{b^8 d^4 (7+p)}+\frac {2 \left (a+b \sqrt {c+d x}\right )^{8+p}}{b^8 d^4 (8+p)} \]
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Rubi [A]
time = 0.20, antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {378, 1412, 786}
\begin {gather*} -\frac {2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{p+1}}{b^8 d^4 (p+1)}+\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+2}}{b^8 d^4 (p+2)}-\frac {6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+3}}{b^8 d^4 (p+3)}-\frac {10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+5}}{b^8 d^4 (p+5)}+\frac {6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+6}}{b^8 d^4 (p+6)}+\frac {2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{p+4}}{b^8 d^4 (p+4)}-\frac {14 a \left (a+b \sqrt {c+d x}\right )^{p+7}}{b^8 d^4 (p+7)}+\frac {2 \left (a+b \sqrt {c+d x}\right )^{p+8}}{b^8 d^4 (p+8)} \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rule 786
Rule 1412
Rubi steps
\begin {align*} \int x^3 \left (a+b \sqrt {c+d x}\right )^p \, dx &=\frac {\text {Subst}\left (\int \left (a+b \sqrt {x}\right )^p (-c+x)^3 \, dx,x,c+d x\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int x (a+b x)^p \left (-c+x^2\right )^3 \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {a \left (a^2-b^2 c\right )^3 (a+b x)^p}{b^7}-\frac {\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2 (a+b x)^{1+p}}{b^7}-\frac {3 \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) (a+b x)^{2+p}}{b^7}+\frac {\left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) (a+b x)^{3+p}}{b^7}-\frac {5 a \left (7 a^2-3 b^2 c\right ) (a+b x)^{4+p}}{b^7}-\frac {3 \left (-7 a^2+b^2 c\right ) (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 {c+d x}\right )}{d^4}\\ &=-\frac {2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{1+p}}{b^8 d^4 (1+p)}+\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{2+p}}{b^8 d^4 (2+p)}-\frac {6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3+p}}{b^8 d^4 (3+p)}+\frac {2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{4+p}}{b^8 d^4 (4+p)}-\frac {10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5+p}}{b^8 d^4 (5+p)}+\frac {6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{6+p}}{b^8 d^4 (6+p)}-\frac {14 a \left (a+b \sqrt {c+d x}\right )^{7+p}}{b^8 d^4 (7+p)}+\frac {2 \left (a+b \sqrt {c+d x}\right )^{8+p}}{b^8 d^4 (8+p)}\\ \end {align*}
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Mathematica [A]
time = 0.79, size = 554, normalized size = 1.58 \begin {gather*} -\frac {2 \left (a+b \sqrt {c+d x}\right )^{1+p} \left (5040 a^7-5040 a^6 b (1+p) \sqrt {c+d x}-360 a^5 b^2 \left (-6 c \left (-7+p+p^2\right )-7 d \left (2+3 p+p^2\right ) x\right )+120 a^4 b^3 (1+p) \sqrt {c+d x} \left (c \left (126+10 p-4 p^2\right )-7 d \left (6+5 p+p^2\right ) x\right )+6 a^3 b^4 \left (8 c^2 \left (315-124 p-139 p^2-14 p^3+p^4\right )+40 c d \left (-42-61 p-16 p^2+4 p^3+p^4\right ) x+35 d^2 \left (24+50 p+35 p^2+10 p^3+p^4\right ) x^2\right )-6 a^2 b^5 (1+p) \sqrt {c+d x} \left (-24 c^2 \left (-105-24 p+5 p^2+p^3\right )+4 c d \left (-420-386 p-94 p^2-p^3+p^4\right ) x+7 d^2 \left (120+154 p+71 p^2+14 p^3+p^4\right ) x^2\right )+b^7 \left (105+176 p+86 p^2+16 p^3+p^4\right ) \sqrt {c+d x} \left (48 c^3-24 c^2 d (2+p) x+6 c d^2 \left (8+6 p+p^2\right ) x^2-d^3 \left (48+44 p+12 p^2+p^3\right ) x^3\right )+a b^6 \left (48 c^3 \left (-105+103 p+138 p^2+38 p^3+3 p^4\right )-24 c^2 d \left (-210-283 p-21 p^2+74 p^3+24 p^4+2 p^5\right ) x+6 c d^2 \left (-840-1726 p-1151 p^2-265 p^3+10 p^4+11 p^5+p^6\right ) x^2+7 d^3 \left (720+1764 p+1624 p^2+735 p^3+175 p^4+21 p^5+p^6\right ) x^3\right )\right )}{b^8 d^4 (1+p) (2+p) (3+p) (4+p) (5+p) (6+p) (7+p) (8+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \sqrt {d x +c}\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 728 vs.
\(2 (335) = 670\).
time = 0.29, size = 728, normalized size = 2.08 \begin {gather*} -\frac {2 \, {\left (\frac {{\left ({\left (d x + c\right )} b^{2} {\left (p + 1\right )} + \sqrt {d x + c} a b p - a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} c^{3}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} - \frac {3 \, {\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} {\left (d x + c\right )}^{2} b^{4} + {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} - 3 \, {\left (p^{2} + p\right )} {\left (d x + c\right )} a^{2} b^{2} + 6 \, \sqrt {d x + c} a^{3} b p - 6 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} c^{2}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} + \frac {3 \, {\left ({\left (p^{5} + 15 \, p^{4} + 85 \, p^{3} + 225 \, p^{2} + 274 \, p + 120\right )} {\left (d x + c\right )}^{3} b^{6} + {\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )} {\left (d x + c\right )}^{\frac {5}{2}} a b^{5} - 5 \, {\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )} {\left (d x + c\right )}^{2} a^{2} b^{4} + 20 \, {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{3} - 60 \, {\left (p^{2} + p\right )} {\left (d x + c\right )} a^{4} b^{2} + 120 \, \sqrt {d x + c} a^{5} b p - 120 \, a^{6}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} c}{{\left (p^{6} + 21 \, p^{5} + 175 \, p^{4} + 735 \, p^{3} + 1624 \, p^{2} + 1764 \, p + 720\right )} b^{6}} - \frac {{\left ({\left (p^{7} + 28 \, p^{6} + 322 \, p^{5} + 1960 \, p^{4} + 6769 \, p^{3} + 13132 \, p^{2} + 13068 \, p + 5040\right )} {\left (d x + c\right )}^{4} b^{8} + {\left (p^{7} + 21 \, p^{6} + 175 \, p^{5} + 735 \, p^{4} + 1624 \, p^{3} + 1764 \, p^{2} + 720 \, p\right )} {\left (d x + c\right )}^{\frac {7}{2}} a b^{7} - 7 \, {\left (p^{6} + 15 \, p^{5} + 85 \, p^{4} + 225 \, p^{3} + 274 \, p^{2} + 120 \, p\right )} {\left (d x + c\right )}^{3} a^{2} b^{6} + 42 \, {\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )} {\left (d x + c\right )}^{\frac {5}{2}} a^{3} b^{5} - 210 \, {\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )} {\left (d x + c\right )}^{2} a^{4} b^{4} + 840 \, {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} {\left (d x + c\right )}^{\frac {3}{2}} a^{5} b^{3} - 2520 \, {\left (p^{2} + p\right )} {\left (d x + c\right )} a^{6} b^{2} + 5040 \, \sqrt {d x + c} a^{7} b p - 5040 \, a^{8}\right )} {\left (\sqrt {d x + c} b + 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}}\right )}}{d^{4}} \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. 1416 vs.
\(2 (335) = 670\).
time = 0.50, size = 1416, normalized size = 4.05 \begin {gather*} -\frac {2 \, {\left (5040 \, b^{8} c^{4} - 20160 \, a^{2} b^{6} c^{3} + 30240 \, a^{4} b^{4} c^{2} - 20160 \, a^{6} b^{2} c + 5040 \, a^{8} + 48 \, {\left (b^{8} c^{4} + 6 \, a^{2} b^{6} c^{3} + a^{4} b^{4} c^{2}\right )} p^{4} - {\left (b^{8} d^{4} p^{7} + 28 \, b^{8} d^{4} p^{6} + 322 \, b^{8} d^{4} p^{5} + 1960 \, b^{8} d^{4} p^{4} + 6769 \, b^{8} d^{4} p^{3} + 13132 \, b^{8} d^{4} p^{2} + 13068 \, b^{8} d^{4} p + 5040 \, b^{8} d^{4}\right )} x^{4} + 384 \, {\left (2 \, b^{8} c^{4} + 7 \, a^{2} b^{6} c^{3} - 3 \, a^{4} b^{4} c^{2}\right )} p^{3} - {\left (b^{8} c d^{3} p^{7} + {\left (22 \, b^{8} c - 7 \, a^{2} b^{6}\right )} d^{3} p^{6} + 5 \, {\left (38 \, b^{8} c - 21 \, a^{2} b^{6}\right )} d^{3} p^{5} + 5 \, {\left (164 \, b^{8} c - 119 \, a^{2} b^{6}\right )} d^{3} p^{4} + {\left (1849 \, b^{8} c - 1575 \, a^{2} b^{6}\right )} d^{3} p^{3} + 2 \, {\left (1019 \, b^{8} c - 959 \, a^{2} b^{6}\right )} d^{3} p^{2} + 840 \, {\left (b^{8} c - a^{2} b^{6}\right )} d^{3} p\right )} x^{3} + 48 \, {\left (86 \, b^{8} c^{4} + 81 \, a^{2} b^{6} c^{3} - 124 \, a^{4} b^{4} c^{2} + 45 \, a^{6} b^{2} c\right )} p^{2} + 6 \, {\left (18 \, b^{8} c^{2} d^{2} p^{5} + {\left (b^{8} c^{2} + a^{2} b^{6} c\right )} d^{2} p^{6} + {\left (118 \, b^{8} c^{2} - 95 \, a^{2} b^{6} c + 35 \, a^{4} b^{4}\right )} d^{2} p^{4} + 6 \, {\left (58 \, b^{8} c^{2} - 80 \, a^{2} b^{6} c + 35 \, a^{4} b^{4}\right )} d^{2} p^{3} + {\left (457 \, b^{8} c^{2} - 806 \, a^{2} b^{6} c + 385 \, a^{4} b^{4}\right )} d^{2} p^{2} + 210 \, {\left (b^{8} c^{2} - 2 \, a^{2} b^{6} c + a^{4} b^{4}\right )} d^{2} p\right )} x^{2} + 192 \, {\left (44 \, b^{8} c^{4} - 71 \, a^{2} b^{6} c^{3} + 54 \, a^{4} b^{4} c^{2} - 15 \, a^{6} b^{2} c\right )} p - 24 \, {\left ({\left (b^{8} c^{3} + 3 \, a^{2} b^{6} c^{2}\right )} d p^{5} + 2 \, {\left (8 \, b^{8} c^{3} + 9 \, a^{2} b^{6} c^{2} - 5 \, a^{4} b^{4} c\right )} d p^{4} + {\left (86 \, b^{8} c^{3} - 57 \, a^{2} b^{6} c^{2} + 15 \, a^{4} b^{4} c\right )} d p^{3} + {\left (176 \, b^{8} c^{3} - 387 \, a^{2} b^{6} c^{2} + 340 \, a^{4} b^{4} c - 105 \, a^{6} b^{2}\right )} d p^{2} + 105 \, {\left (b^{8} c^{3} - 3 \, a^{2} b^{6} c^{2} + 3 \, a^{4} b^{4} c - a^{6} b^{2}\right )} d p\right )} x + {\left (192 \, {\left (a b^{7} c^{3} + a^{3} b^{5} c^{2}\right )} p^{4} + 96 \, {\left (27 \, a b^{7} c^{3} + 2 \, a^{3} b^{5} c^{2} - 5 \, a^{5} b^{3} c\right )} p^{3} - {\left (a b^{7} d^{3} p^{7} + 21 \, a b^{7} d^{3} p^{6} + 175 \, a b^{7} d^{3} p^{5} + 735 \, a b^{7} d^{3} p^{4} + 1624 \, a b^{7} d^{3} p^{3} + 1764 \, a b^{7} d^{3} p^{2} + 720 \, a b^{7} d^{3} p\right )} x^{3} + 192 \, {\left (56 \, a b^{7} c^{3} - 49 \, a^{3} b^{5} c^{2} + 15 \, a^{5} b^{3} c\right )} p^{2} + 6 \, {\left (2 \, a b^{7} c d^{2} p^{6} + {\left (33 \, a b^{7} c - 7 \, a^{3} b^{5}\right )} d^{2} p^{5} + 10 \, {\left (20 \, a b^{7} c - 7 \, a^{3} b^{5}\right )} d^{2} p^{4} + 5 \, {\left (111 \, a b^{7} c - 49 \, a^{3} b^{5}\right )} d^{2} p^{3} + 2 \, {\left (349 \, a b^{7} c - 175 \, a^{3} b^{5}\right )} d^{2} p^{2} + 24 \, {\left (13 \, a b^{7} c - 7 \, a^{3} b^{5}\right )} d^{2} p\right )} x^{2} + 48 \, {\left (279 \, a b^{7} c^{3} - 511 \, a^{3} b^{5} c^{2} + 385 \, a^{5} b^{3} c - 105 \, a^{7} b\right )} p - 24 \, {\left ({\left (3 \, a b^{7} c^{2} + a^{3} b^{5} c\right )} d p^{5} + 2 \, {\left (21 \, a b^{7} c^{2} - 5 \, a^{3} b^{5} c\right )} d p^{4} + {\left (192 \, a b^{7} c^{2} - 135 \, a^{3} b^{5} c + 35 \, a^{5} b^{3}\right )} d p^{3} + {\left (327 \, a b^{7} c^{2} - 320 \, a^{3} b^{5} c + 105 \, a^{5} b^{3}\right )} d p^{2} + 2 \, {\left (87 \, a b^{7} c^{2} - 98 \, a^{3} b^{5} c + 35 \, a^{5} b^{3}\right )} d p\right )} x\right )} \sqrt {d x + c}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p}}{b^{8} d^{4} p^{8} + 36 \, b^{8} d^{4} p^{7} + 546 \, b^{8} d^{4} p^{6} + 4536 \, b^{8} d^{4} p^{5} + 22449 \, b^{8} d^{4} p^{4} + 67284 \, b^{8} d^{4} p^{3} + 118124 \, b^{8} d^{4} p^{2} + 109584 \, b^{8} d^{4} p + 40320 \, b^{8} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \sqrt {c + d x}\right )^{p}\, dx \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. 5699 vs.
\(2 (335) = 670\).
time = 4.13, size = 5699, normalized size = 16.28 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\left (a+b\,\sqrt {c+d\,x}\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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