Optimal. Leaf size=324 \[ \frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac {4 a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4} \]
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Rubi [A] time = 0.23, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {371, 1398, 772} \[ \frac {4 \left (-30 a^2 b^2 c+35 a^4+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac {4 a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4} \]
Antiderivative was successfully verified.
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Rule 371
Rule 772
Rule 1398
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-c+x)^3}{\sqrt {a+b \sqrt {x}}} \, dx,x,c+d x\right )}{d^4}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-c+x^2\right )^3}{\sqrt {a+b x}} \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {a \left (a^2-b^2 c\right )^3}{b^7 \sqrt {a+b x}}-\frac {\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2 \sqrt {a+b x}}{b^7}-\frac {3 \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) (a+b x)^{3/2}}{b^7}+\frac {\left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) (a+b x)^{5/2}}{b^7}-\frac {5 a \left (7 a^2-3 b^2 c\right ) (a+b x)^{7/2}}{b^7}-\frac {3 \left (-7 a^2+b^2 c\right ) (a+b x)^{9/2}}{b^7}-\frac {7 a (a+b x)^{11/2}}{b^7}+\frac {(a+b x)^{13/2}}{b^7}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=-\frac {4 a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac {12 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 )^{5/2}}{5 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 232, normalized size = 0.72 \[ \frac {4 \sqrt {a+b \sqrt {c+d x}} \left (-14336 a^7+7168 a^6 b \sqrt {c+d x}+768 a^5 b^2 (58 c-7 d x)-640 a^4 b^3 (32 c-7 d x) \sqrt {c+d x}-16 a^3 b^4 \left (2936 c^2-680 c d x+245 d^2 x^2\right )+24 a^2 b^5 \sqrt {c+d x} \left (784 c^2-356 c d x+147 d^2 x^2\right )+6 a b^6 \left (2880 c^3-928 c^2 d x+658 c d^2 x^2-539 d^3 x^3\right )-39 b^7 \sqrt {c+d x} \left (128 c^3-96 c^2 d x+84 c d^2 x^2-77 d^3 x^3\right )\right )}{45045 b^8 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 231, normalized size = 0.71 \[ -\frac {4 \, {\left (3234 \, a b^{6} d^{3} x^{3} - 17280 \, a b^{6} c^{3} + 46976 \, a^{3} b^{4} c^{2} - 44544 \, a^{5} b^{2} c + 14336 \, a^{7} - 28 \, {\left (141 \, a b^{6} c - 140 \, a^{3} b^{4}\right )} d^{2} x^{2} + 64 \, {\left (87 \, a b^{6} c^{2} - 170 \, a^{3} b^{4} c + 84 \, a^{5} b^{2}\right )} d x - {\left (3003 \, b^{7} d^{3} x^{3} - 4992 \, b^{7} c^{3} + 18816 \, a^{2} b^{5} c^{2} - 20480 \, a^{4} b^{3} c + 7168 \, a^{6} b - 252 \, {\left (13 \, b^{7} c - 14 \, a^{2} b^{5}\right )} d^{2} x^{2} + 32 \, {\left (117 \, b^{7} c^{2} - 267 \, a^{2} b^{5} c + 140 \, a^{4} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{45045 \, b^{8} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 409, normalized size = 1.26 \[ -\frac {4 \, {\left (15015 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{6} c^{3} - 45045 \, \sqrt {\sqrt {d x + c} b + a} a b^{6} c^{3} - 19305 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{4} c^{2} + 81081 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{4} c^{2} - 135135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{4} c^{2} + 135135 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{4} c^{2} + 12285 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} b^{2} c - 75075 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a b^{2} c + 193050 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} b^{2} c - 270270 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} b^{2} c + 225225 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} b^{2} c - 135135 \, \sqrt {\sqrt {d x + c} b + a} a^{5} b^{2} c - 3003 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} + 24255 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} a - 85995 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a^{2} + 175175 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{3} - 225225 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{4} + 189189 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{5} - 105105 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{6} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{7}\right )}}{45045 \, b^{8} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 383, normalized size = 1.18 \[ \frac {-\frac {28 \left (a +\sqrt {d x +c}\, b \right )^{\frac {13}{2}} a}{13}-4 \left (-b^{2} c +a^{2}\right )^{3} \sqrt {a +\sqrt {d x +c}\, b}\, a +\frac {4 \left (a +\sqrt {d x +c}\, b \right )^{\frac {15}{2}}}{15}+\frac {4 \left (-3 b^{2} c +21 a^{2}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {11}{2}}}{11}+\frac {4 \left (-8 \left (-b^{2} c +a^{2}\right ) a -2 \left (-2 b^{2} c +6 a^{2}\right ) a -\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {9}{2}}}{9}+\frac {4 \left (8 \left (-b^{2} c +a^{2}\right ) a^{2}-\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 \left (-2 b^{2} c +6 a^{2}\right ) a \right ) a +\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {7}{2}}}{7}+\frac {4 \left (-6 \left (-b^{2} c +a^{2}\right )^{2} a -\left (8 \left (-b^{2} c +a^{2}\right ) a^{2}+\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {5}{2}}}{5}+\frac {4 \left (6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}+\left (-b^{2} c +a^{2}\right )^{3}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}}}{3}}{b^{8} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 268, normalized size = 0.83 \[ \frac {4 \, {\left (3003 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} - 24255 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} a - 12285 \, {\left (b^{2} c - 7 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} + 25025 \, {\left (3 \, a b^{2} c - 7 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} + 6435 \, {\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} - 27027 \, {\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} - 15015 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} + 45045 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \sqrt {\sqrt {d x + c} b + a}\right )}}{45045 \, b^{8} d^{4}} \]
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 \[ \int \frac {x^3}{\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {a + b \sqrt {c + d x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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