Optimal. Leaf size=133 \[ -\frac {4 a \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^4 d^2}+\frac {4 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^4 d^2}-\frac {12 a \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^4 d^2}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^4 d^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 133, 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 {4 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^4 d^2}-\frac {4 a \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^4 d^2}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^4 d^2}-\frac {12 a \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rule 786
Rule 1412
Rubi steps
\begin {align*} \int x \sqrt {a+b \sqrt {c+d x}} \, dx &=\frac {\text {Subst}\left (\int \sqrt {a+b \sqrt {x}} (-c+x) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 \text {Subst}\left (\int x \sqrt {a+b x} \left (-c+x^2\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {2 \text {Subst}\left (\int \left (\frac {\left (-a^3+a b^2 c\right ) \sqrt {a+b x}}{b^3}+\frac {\left (3 a^2-b^2 c\right ) (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {4 a \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^4 d^2}+\frac {4 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^4 d^2}-\frac {12 a \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^4 d^2}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 84, normalized size = 0.63 \begin {gather*} \frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (-16 a^3+6 a b^2 (2 c-5 d x)+24 a^2 b \sqrt {c+d x}+7 b^3 \sqrt {c+d x} (-4 c+5 d x)\right )}{315 b^4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 94, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}-\frac {12 a \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {4 \left (-b^{2} c +3 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-b^{2} c +a^{2}\right ) a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{b^{4} d^{2}}\) | \(94\) |
default | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}-\frac {12 a \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {4 \left (-b^{2} c +3 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-b^{2} c +a^{2}\right ) a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{b^{4} d^{2}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 93, normalized size = 0.70 \begin {gather*} \frac {4 \, {\left (35 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} - 135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a - 63 \, {\left (b^{2} c - 3 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} + 105 \, {\left (a b^{2} c - a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}\right )}}{315 \, b^{4} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 103, normalized size = 0.77 \begin {gather*} \frac {4 \, {\left (35 \, b^{4} d^{2} x^{2} - 28 \, b^{4} c^{2} + 36 \, a^{2} b^{2} c - 16 \, a^{4} + {\left (7 \, b^{4} c - 6 \, a^{2} b^{2}\right )} d x + {\left (5 \, a b^{3} d x - 16 \, a b^{3} c + 8 \, a^{3} b\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{315 \, b^{4} d^{2}} \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 \sqrt {a + b \sqrt {c + d x}}\, 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. 279 vs.
\(2 (109) = 218\).
time = 3.62, size = 279, normalized size = 2.10 \begin {gather*} -\frac {4 \, {\left (\frac {3 \, {\left (35 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{2} c - 105 \, \sqrt {\sqrt {d x + c} b + a} a b^{2} c - 15 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} + 63 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a - 105 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} + 105 \, \sqrt {\sqrt {d x + c} b + a} a^{3}\right )} a}{b^{3} d} + \frac {63 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} b^{2} c - 210 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a b^{2} c + 315 \, \sqrt {\sqrt {d x + c} b + a} a^{2} b^{2} c - 35 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} + 180 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a - 378 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{2} + 420 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{3} - 315 \, \sqrt {\sqrt {d x + c} b + a} a^{4}}{b^{3} d}\right )}}{315 \, b d} \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 = -0.01 \begin {gather*} \int x\,\sqrt {a+b\,\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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