Optimal. Leaf size=142 \[ -\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}+\frac {5 c^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214}
\begin {gather*} \frac {5 c^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2}}-\frac {5 c \sqrt {c+d x} (b c-a d)}{a^3 \sqrt {a+b x}}-\frac {5 (c+d x)^{3/2} (b c-a d)}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx &=-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}-\frac {(5 (b c-a d)) \int \frac {(c+d x)^{3/2}}{x (a+b x)^{5/2}} \, dx}{2 a}\\ &=-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}-\frac {(5 c (b c-a d)) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{2 a^2}\\ &=-\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}-\frac {\left (5 c^2 (b c-a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^3}\\ &=-\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}-\frac {\left (5 c^2 (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a^3}\\ &=-\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}+\frac {5 c^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.56, size = 171, normalized size = 1.20 \begin {gather*} \frac {\sqrt {c+d x} \left (-15 b^2 c^2 x^2+10 a b c x (-2 c+d x)+a^2 \left (-3 c^2+14 c d x+2 d^2 x^2\right )\right )}{3 a^3 x (a+b x)^{3/2}}+\frac {5 c^{3/2} \sqrt {\frac {b}{d}} \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \left (-b x+\sqrt {\frac {b}{d}} \sqrt {a+b x} \sqrt {c+d x}\right )}{\sqrt {a} \sqrt {b} \sqrt {c}}\right )}{a^{7/2} \sqrt {b}} \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. \(501\) vs.
\(2(116)=232\).
time = 0.08, size = 502, normalized size = 3.54
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c^{2} d x -15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} x -4 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}-20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}+30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}-28 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x +40 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x +6 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{6 a^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}}}\) | \(502\) |
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. 237 vs.
\(2 (116) = 232\).
time = 1.22, size = 505, normalized size = 3.56 \begin {gather*} \left [-\frac {15 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \, {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} + {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a^{2} c^{2} + {\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}, -\frac {15 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \, {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} + {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + 2 \, {\left (3 \, a^{2} c^{2} + {\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}\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. 990 vs.
\(2 (116) = 232\).
time = 4.22, size = 990, normalized size = 6.97 \begin {gather*} \frac {5 \, {\left (\sqrt {b d} b c^{3} {\left | b \right |} - \sqrt {b d} a c^{2} d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a^{3} b} - \frac {2 \, {\left (\sqrt {b d} b^{3} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c^{3} d {\left | b \right |} + \sqrt {b d} a^{2} b c^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b c^{3} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a c^{2} d {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a^{3}} - \frac {4 \, {\left (6 \, \sqrt {b d} b^{7} c^{5} {\left | b \right |} - 23 \, \sqrt {b d} a b^{6} c^{4} d {\left | b \right |} + 32 \, \sqrt {b d} a^{2} b^{5} c^{3} d^{2} {\left | b \right |} - 18 \, \sqrt {b d} a^{3} b^{4} c^{2} d^{3} {\left | b \right |} + 2 \, \sqrt {b d} a^{4} b^{3} c d^{4} {\left | b \right |} + \sqrt {b d} a^{5} b^{2} d^{5} {\left | b \right |} - 12 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c^{4} {\left | b \right |} + 36 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} c^{3} d {\left | b \right |} - 36 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} c^{2} d^{2} {\left | b \right |} + 12 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} c d^{3} {\left | b \right |} + 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} c^{3} {\left | b \right |} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{2} c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} d^{3} {\left | b \right |}\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} a^{3} b^{2}} \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 \frac {{\left (c+d\,x\right )}^{5/2}}{x^2\,{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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