Optimal. Leaf size=101 \[ -\frac {2 (c+d x)^{7/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {8 d (c+d x)^{7/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{7/2}}{693 (b c-a d)^3 (a+b x)^{7/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} -\frac {16 d^2 (c+d x)^{7/2}}{693 (a+b x)^{7/2} (b c-a d)^3}+\frac {8 d (c+d x)^{7/2}}{99 (a+b x)^{9/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx &=-\frac {2 (c+d x)^{7/2}}{11 (b c-a d) (a+b x)^{11/2}}-\frac {(4 d) \int \frac {(c+d x)^{5/2}}{(a+b x)^{11/2}} \, dx}{11 (b c-a d)}\\ &=-\frac {2 (c+d x)^{7/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {8 d (c+d x)^{7/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}+\frac {\left (8 d^2\right ) \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx}{99 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{7/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {8 d (c+d x)^{7/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{7/2}}{693 (b c-a d)^3 (a+b x)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 73, normalized size = 0.72 \begin {gather*} -\frac {2 (c+d x)^{7/2} \left (99 d^2-\frac {154 b d (c+d x)}{a+b x}+\frac {63 b^2 (c+d x)^2}{(a+b x)^2}\right )}{693 (b c-a d)^3 (a+b x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs.
\(2(83)=166\).
time = 0.16, size = 314, normalized size = 3.11
method | result | size |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (8 b^{2} x^{2} d^{2}+44 a b \,d^{2} x -28 b^{2} c d x +99 a^{2} d^{2}-154 a b c d +63 b^{2} c^{2}\right )}{693 \left (b x +a \right )^{\frac {11}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
default | \(-\frac {\left (d x +c \right )^{\frac {5}{2}}}{3 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {5 \left (a d -b c \right ) \left (-\frac {\left (d x +c \right )^{\frac {3}{2}}}{4 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{5 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{10 b}\right )}{8 b}\right )}{6 b}\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 513 vs.
\(2 (83) = 166\).
time = 10.56, size = 513, normalized size = 5.08 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} d^{5} x^{5} + 63 \, b^{2} c^{5} - 154 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} - 4 \, {\left (b^{2} c d^{4} - 11 \, a b d^{5}\right )} x^{4} + {\left (3 \, b^{2} c^{2} d^{3} - 22 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{3} + {\left (113 \, b^{2} c^{3} d^{2} - 330 \, a b c^{2} d^{3} + 297 \, a^{2} c d^{4}\right )} x^{2} + {\left (161 \, b^{2} c^{4} d - 418 \, a b c^{3} d^{2} + 297 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{693 \, {\left (a^{6} b^{3} c^{3} - 3 \, a^{7} b^{2} c^{2} d + 3 \, a^{8} b c d^{2} - a^{9} d^{3} + {\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} x^{6} + 6 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} x^{5} + 15 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} x^{4} + 20 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} x^{3} + 15 \, {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} x^{2} + 6 \, {\left (a^{5} b^{4} c^{3} - 3 \, a^{6} b^{3} c^{2} d + 3 \, a^{7} b^{2} c d^{2} - a^{8} b d^{3}\right )} x\right )}} \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. 2316 vs.
\(2 (83) = 166\).
time = 1.91, size = 2316, normalized size = 22.93 \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 = 1.35, size = 333, normalized size = 3.30 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {198\,a^2\,c^3\,d^2-308\,a\,b\,c^4\,d+126\,b^2\,c^5}{693\,b^5\,{\left (a\,d-b\,c\right )}^3}+\frac {x^3\,\left (198\,a^2\,d^5-44\,a\,b\,c\,d^4+6\,b^2\,c^2\,d^3\right )}{693\,b^5\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,d^5\,x^5}{693\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{693\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,c\,d^2\,x^2\,\left (297\,a^2\,d^2-330\,a\,b\,c\,d+113\,b^2\,c^2\right )}{693\,b^5\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,c^2\,d\,x\,\left (297\,a^2\,d^2-418\,a\,b\,c\,d+161\,b^2\,c^2\right )}{693\,b^5\,{\left (a\,d-b\,c\right )}^3}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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