Optimal. Leaf size=175 \[ -\frac {3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{7/2}}+\frac {3 \sqrt {a+b x} (b c-5 a d) (b c-a d)}{4 a c^3 \sqrt {c+d x}}-\frac {(a+b x)^{3/2} (b c-5 a d)}{4 a c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 a c x^2 \sqrt {c+d x}} \]
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Rubi [A] time = 0.08, antiderivative size = 175, 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 = {96, 94, 93, 208} \[ -\frac {(a+b x)^{3/2} (b c-5 a d)}{4 a c^2 x \sqrt {c+d x}}+\frac {3 \sqrt {a+b x} (b c-5 a d) (b c-a d)}{4 a c^3 \sqrt {c+d x}}-\frac {3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{7/2}}-\frac {(a+b x)^{5/2}}{2 a c x^2 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{3/2}} \, dx &=-\frac {(a+b x)^{5/2}}{2 a c x^2 \sqrt {c+d x}}-\frac {\left (-\frac {b c}{2}+\frac {5 a d}{2}\right ) \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx}{2 a c}\\ &=-\frac {(b c-5 a d) (a+b x)^{3/2}}{4 a c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 (b c-5 a d) (b c-a d)) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{8 a c^2}\\ &=\frac {3 (b c-5 a d) (b c-a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {(b c-5 a d) (a+b x)^{3/2}}{4 a c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 (b c-5 a d) (b c-a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 c^3}\\ &=\frac {3 (b c-5 a d) (b c-a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {(b c-5 a d) (a+b x)^{3/2}}{4 a c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 (b c-5 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 c^3}\\ &=\frac {3 (b c-5 a d) (b c-a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {(b c-5 a d) (a+b x)^{3/2}}{4 a c^2 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2}}{2 a c x^2 \sqrt {c+d x}}-\frac {3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 130, normalized size = 0.74 \[ \frac {\sqrt {a+b x} \left (a \left (-2 c^2+5 c d x+15 d^2 x^2\right )-b c x (5 c+13 d x)\right )}{4 c^3 x^2 \sqrt {c+d x}}-\frac {3 \left (5 a^2 d^2-6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.00, size = 472, normalized size = 2.70 \[ \left [\frac {3 \, {\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{3} + {\left (13 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a c^{4} d x^{3} + a c^{5} x^{2}\right )}}, \frac {3 \, {\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{3} + {\left (13 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a c^{4} d x^{3} + a c^{5} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 15.57, size = 1096, normalized size = 6.26 \[ -\frac {2 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c^{3} {\left | b \right |}} - \frac {3 \, {\left (\sqrt {b d} b^{4} c^{2} - 6 \, \sqrt {b d} a b^{3} c d + 5 \, \sqrt {b d} a^{2} b^{2} d^{2}\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} b c^{3} {\left | b \right |}} - \frac {5 \, \sqrt {b d} b^{10} c^{5} - 27 \, \sqrt {b d} a b^{9} c^{4} d + 58 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} - 62 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} + 33 \, \sqrt {b d} a^{4} b^{6} c d^{4} - 7 \, \sqrt {b d} a^{5} b^{5} d^{5} - 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} b^{8} c^{4} + 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 b^{7} c^{3} d - 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^{6} c^{2} d^{2} - 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^{3} b^{5} c d^{3} + 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} a^{4} b^{4} d^{4} + 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} b^{6} c^{3} - 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 b^{5} c^{2} d + \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^{4} c d^{2} - 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} a^{3} b^{3} d^{3} - 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} b^{4} c^{2} - 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^{3} c d + 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} a^{2} b^{2} d^{2}}{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} c^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 464, normalized size = 2.65 \[ -\frac {\sqrt {b x +a}\, \left (15 a^{2} d^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-18 a b c \,d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 b^{2} c^{2} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+15 a^{2} c \,d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-18 a b \,c^{2} d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 b^{2} c^{3} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,d^{2} x^{2}+26 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c d \,x^{2}-10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c d x +10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2} x +4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,c^{2}\right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, \sqrt {d x +c}\, c^{3} x^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{3/2}}{x^3\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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