Optimal. Leaf size=124 \[ -\frac {5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {5 b d}{(b c-a d)^3 \sqrt {c+d x}}+\frac {5 b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{7/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {44, 53, 65, 214}
\begin {gather*} \frac {5 b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac {5 b d}{\sqrt {c+d x} (b c-a d)^3}-\frac {1}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac {5 d}{3 (c+d x)^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^2 (c+d x)^{5/2}} \, dx &=-\frac {1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {(5 d) \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx}{2 (b c-a d)}\\ &=-\frac {5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {(5 b d) \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx}{2 (b c-a d)^2}\\ &=-\frac {5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {5 b d}{(b c-a d)^3 \sqrt {c+d x}}-\frac {\left (5 b^2 d\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 (b c-a d)^3}\\ &=-\frac {5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {5 b d}{(b c-a d)^3 \sqrt {c+d x}}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{(b c-a d)^3}\\ &=-\frac {5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {5 b d}{(b c-a d)^3 \sqrt {c+d x}}+\frac {5 b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 125, normalized size = 1.01 \begin {gather*} \frac {2 a^2 d^2-2 a b d (7 c+5 d x)-b^2 \left (3 c^2+20 c d x+15 d^2 x^2\right )}{3 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}+\frac {5 b^{3/2} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}} \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 = 0.17, size = 121, normalized size = 0.98
method | result | size |
derivativedivides | \(2 d \left (-\frac {1}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b}{\left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {b^{2} \left (\frac {\sqrt {d x +c}}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3}}\right )\) | \(121\) |
default | \(2 d \left (-\frac {1}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b}{\left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {b^{2} \left (\frac {\sqrt {d x +c}}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3}}\right )\) | \(121\) |
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. 386 vs.
\(2 (106) = 212\).
time = 0.33, size = 782, normalized size = 6.31 \begin {gather*} \left [-\frac {15 \, {\left (b^{2} d^{3} x^{3} + a b c^{2} d + {\left (2 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2} + {\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) + 2 \, {\left (15 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 14 \, a b c d - 2 \, a^{2} d^{2} + 10 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {d x + c}}{6 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}}, \frac {15 \, {\left (b^{2} d^{3} x^{3} + a b c^{2} d + {\left (2 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2} + {\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) - {\left (15 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 14 \, a b c d - 2 \, a^{2} d^{2} + 10 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {d x + c}}{3 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}}\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}{\left (a + b x\right )^{2} \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. 216 vs.
\(2 (106) = 212\).
time = 0.00, size = 241, normalized size = 1.94 \begin {gather*} 2 \left (\frac {\sqrt {c+d x} b^{2} d}{\left (2 b^{3} c^{3}-6 b^{2} d c^{2} a+6 b d^{2} c a^{2}-2 d^{3} a^{3}\right ) \left (-\left (c+d x\right ) b+b c-d a\right )}+\frac {6 \left (c+d x\right ) b d+b d c-d^{2} a}{\left (-3 b^{3} c^{3}+9 b^{2} d c^{2} a-9 b d^{2} c a^{2}+3 d^{3} a^{3}\right ) \sqrt {c+d x} \left (c+d x\right )}+\frac {5 b^{2} d \arctan \left (\frac {b \sqrt {c+d x}}{\sqrt {-b^{2} c+a b d}}\right )}{2 \left (-b^{3} c^{3}+3 b^{2} d c^{2} a-3 b d^{2} c a^{2}+d^{3} a^{3}\right ) \sqrt {-b^{2} c+a b d}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 161, normalized size = 1.30 \begin {gather*} \frac {\frac {10\,b\,d\,\left (c+d\,x\right )}{3\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,d}{3\,\left (a\,d-b\,c\right )}+\frac {5\,b^2\,d\,{\left (c+d\,x\right )}^2}{{\left (a\,d-b\,c\right )}^3}}{b\,{\left (c+d\,x\right )}^{5/2}+\left (a\,d-b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}+\frac {5\,b^{3/2}\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^{7/2}}\right )}{{\left (a\,d-b\,c\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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