Optimal. Leaf size=66 \[ \frac {2 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/2}}+\frac {4 b \sqrt {a+b x}}{3 (b c-a d)^2 \sqrt {c+d x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 {4 b \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^2}+\frac {2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx &=\frac {2 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/2}}+\frac {(2 b) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)}\\ &=\frac {2 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/2}}+\frac {4 b \sqrt {a+b x}}{3 (b c-a d)^2 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 46, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {a+b x} (3 b c-a d+2 b d x)}{3 (b c-a d)^2 (c+d x)^{3/2}} \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 [A]
time = 0.16, size = 55, normalized size = 0.83
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-2 b d x +a d -3 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(53\) |
default | \(-\frac {2 \sqrt {b x +a}}{3 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 \left (a d -b c \right )^{2} \sqrt {d x +c}}\) | \(55\) |
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. 118 vs.
\(2 (54) = 108\).
time = 0.37, size = 118, normalized size = 1.79 \begin {gather*} \frac {2 \, {\left (2 \, b d x + 3 \, b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \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 \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, 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. 126 vs.
\(2 (54) = 108\).
time = 0.02, size = 174, normalized size = 2.64 \begin {gather*} \frac {4 \left (-\frac {6 b^{4} d^{2} \sqrt {a+b x} \sqrt {a+b x}}{-18 b^{2} d c^{2} \left |b\right |+36 b d^{2} a c \left |b\right |-18 d^{3} a^{2} \left |b\right |}-\frac {9 b^{5} d c-9 b^{4} d^{2} a}{-18 b^{2} d c^{2} \left |b\right |+36 b d^{2} a c \left |b\right |-18 d^{3} a^{2} \left |b\right |}\right ) \sqrt {a+b x} \sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}}{\left (-a b d+b^{2} c+b d \left (a+b x\right )\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 127, normalized size = 1.92 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (6\,c\,b^2+2\,a\,d\,b\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,a^2\,d-6\,a\,b\,c}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,b^2\,x^2}{3\,d\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,\sqrt {a+b\,x}+\frac {c^2\,\sqrt {a+b\,x}}{d^2}+\frac {2\,c\,x\,\sqrt {a+b\,x}}{d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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