Optimal. Leaf size=178 \[ \frac {3 (b c-a d) (5 b c-a d) \sqrt {c+d x}}{4 a^3 c \sqrt {a+b x}}+\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}-\frac {3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} \sqrt {c}} \]
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Rubi [A]
time = 0.05, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {98, 96, 95, 214}
\begin {gather*} -\frac {3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} \sqrt {c}}+\frac {3 \sqrt {c+d x} (b c-a d) (5 b c-a d)}{4 a^3 c \sqrt {a+b x}}+\frac {(c+d x)^{3/2} (5 b c-a d)}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx &=-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}-\frac {\left (\frac {5 b c}{2}-\frac {a d}{2}\right ) \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx}{2 a c}\\ &=\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}+\frac {(3 (b c-a d) (5 b c-a d)) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^2 c}\\ &=\frac {3 (b c-a d) (5 b c-a d) \sqrt {c+d x}}{4 a^3 c \sqrt {a+b x}}+\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}+\frac {(3 (b c-a d) (5 b c-a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^3}\\ &=\frac {3 (b c-a d) (5 b c-a d) \sqrt {c+d x}}{4 a^3 c \sqrt {a+b x}}+\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}+\frac {(3 (b c-a d) (5 b c-a d)) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^3}\\ &=\frac {3 (b c-a d) (5 b c-a d) \sqrt {c+d x}}{4 a^3 c \sqrt {a+b x}}+\frac {(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a c x^2 \sqrt {a+b x}}-\frac {3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 1.68, size = 174, normalized size = 0.98 \begin {gather*} \frac {\sqrt {c+d x} \left (15 b^2 c x^2+a b x (5 c-13 d x)-a^2 (2 c+5 d x)\right )}{4 a^3 x^2 \sqrt {a+b x}}-\frac {3 \sqrt {\frac {b}{d}} \sqrt {d} \left (5 b^2 c^2-6 a b c d+a^2 d^2\right ) \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 )}{4 a^{7/2} \sqrt {b} \sqrt {c}} \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. \(463\) vs.
\(2(146)=292\).
time = 0.07, size = 464, normalized size = 2.61
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,d^{2} x^{3}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{2} x^{3}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{2} x^{2}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} x^{2}+26 a b d \,x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-30 b^{2} c \,x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+10 a^{2} d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-10 a b c x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+4 a^{2} c \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{8 a^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{2} \sqrt {a c}\, \sqrt {b x +a}}\) | \(464\) |
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 [A]
time = 2.49, size = 474, normalized size = 2.66 \begin {gather*} \left [\frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {a c} \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 c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{3} c^{2} - {\left (15 \, a b^{2} c^{2} - 13 \, a^{2} b c d\right )} x^{2} - 5 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a^{4} b c x^{3} + a^{5} c x^{2}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{3} c^{2} - {\left (15 \, a b^{2} c^{2} - 13 \, a^{2} b c d\right )} x^{2} - 5 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a^{4} b c x^{3} + a^{5} c x^{2}\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 {\left (c + d x\right )^{\frac {3}{2}}}{x^{3} \left (a + b x\right )^{\frac {3}{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. 1167 vs.
\(2 (146) = 292\).
time = 4.59, size = 1167, normalized size = 6.56 \begin {gather*} -\frac {3 \, {\left (5 \, \sqrt {b d} b^{2} c^{2} {\left | b \right |} - 6 \, \sqrt {b d} a b c d {\left | b \right |} + \sqrt {b d} a^{2} d^{2} {\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 )}{4 \, \sqrt {-a b c d} a^{3} b} + \frac {4 \, {\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a b c d {\left | b \right |} + \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )}}{{\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 )} a^{3}} + \frac {7 \, \sqrt {b d} b^{8} c^{5} {\left | b \right |} - 33 \, \sqrt {b d} a b^{7} c^{4} d {\left | b \right |} + 62 \, \sqrt {b d} a^{2} b^{6} c^{3} d^{2} {\left | b \right |} - 58 \, \sqrt {b d} a^{3} b^{5} c^{2} d^{3} {\left | b \right |} + 27 \, \sqrt {b d} a^{4} b^{4} c d^{4} {\left | b \right |} - 5 \, \sqrt {b d} a^{5} b^{3} d^{5} {\left | b \right |} - 21 \, \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^{6} c^{4} {\left | b \right |} + 32 \, \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^{5} c^{3} d {\left | b \right |} + 14 \, \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^{4} c^{2} d^{2} {\left | b \right |} - 40 \, \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^{3} c d^{3} {\left | b \right |} + 15 \, \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^{4} b^{2} d^{4} {\left | b \right |} + 21 \, \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^{4} 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 )}^{4} a b^{3} c^{2} d {\left | b \right |} + 11 \, \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^{2} b^{2} c d^{2} {\left | b \right |} - 15 \, \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} b d^{3} {\left | b \right |} - 7 \, \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 )}^{6} b^{2} c^{2} {\left | b \right |} + 2 \, \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 )}^{6} a b c d {\left | b \right |} + 5 \, \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 )}^{6} a^{2} d^{2} {\left | b \right |}}{2 \, {\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 )}^{2} a^{3}} \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 )}^{3/2}}{x^3\,{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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