Optimal. Leaf size=32 \[ -\frac {2 (c+d x)^{5/2}}{5 (b c-a d) (a+b x)^{5/2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37}
\begin {gather*} -\frac {2 (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{5 (b c-a d) (a+b x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 (c+d x)^{5/2}}{5 (b c-a d) (a+b x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(160\) vs.
\(2(26)=52\).
time = 0.18, size = 161, normalized size = 5.03
method | result | size |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 \left (b x +a \right )^{\frac {5}{2}} \left (a d -b c \right )}\) | \(27\) |
default | \(-\frac {\left (d x +c \right )^{\frac {3}{2}}}{b \left (b x +a \right )^{\frac {5}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{2 b \left (b x +a \right )^{\frac {5}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{4 b}\right )}{2 b}\) | \(161\) |
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. 104 vs.
\(2 (26) = 52\).
time = 0.47, size = 104, normalized size = 3.25 \begin {gather*} -\frac {2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{5 \, {\left (a^{3} b c - a^{4} d + {\left (b^{4} c - a b^{3} d\right )} x^{3} + 3 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2} + 3 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)]
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. 82 vs.
\(2 (26) = 52\).
time = 0.08, size = 149, normalized size = 4.66 \begin {gather*} -\frac {2 \left (45 b^{3} d^{6} c-45 b^{2} d^{7} a\right ) \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {c+d x} \sqrt {a d^{2}-b c d+b d \left (c+d x\right )}}{\left (225 b^{4} c^{2} \left |d\right |-450 b^{3} d a c \left |d\right |+225 b^{2} d^{2} a^{2} \left |d\right |\right ) \left (a d^{2}-b c d+b d \left (c+d x\right )\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 27, normalized size = 0.84 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{5/2}}{\left (5\,a\,d-5\,b\,c\right )\,{\left (a+b\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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