Optimal. Leaf size=124 \[ \frac {(3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}-\frac {d \sqrt {a+b x} (b c-3 a d)}{a c^2 \sqrt {c+d x} (b c-a d)}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}} \]
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Rubi [A] time = 0.08, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {103, 152, 12, 93, 208} \[ \frac {(3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}-\frac {d \sqrt {a+b x} (b c-3 a d)}{a c^2 \sqrt {c+d x} (b c-a d)}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 152
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx &=-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}-\frac {\int \frac {\frac {1}{2} (b c+3 a d)+b d x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{a c}\\ &=-\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}+\frac {2 \int -\frac {(b c-a d) (b c+3 a d)}{4 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a c^2 (b c-a d)}\\ &=-\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}-\frac {(b c+3 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a c^2}\\ &=-\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}-\frac {(b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a c^2}\\ &=-\frac {d (b c-3 a d) \sqrt {a+b x}}{a c^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{a c x \sqrt {c+d x}}+\frac {(b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 126, normalized size = 1.02 \[ \frac {\left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} (a d (c+3 d x)-b c (c+d x))}{x \sqrt {c+d x}}}{a^{3/2} c^{5/2} (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.61, size = 492, normalized size = 3.97 \[ \left [\frac {{\left ({\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{2} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x\right )}}, -\frac {{\left ({\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{2} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.99, size = 476, normalized size = 3.84 \[ \frac {2 \, \sqrt {b x + a} b^{2} d^{2}}{{\left (b c^{3} {\left | b \right |} - a c^{2} d {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (\sqrt {b d} b^{3} c + 3 \, \sqrt {b d} a b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b c^{2} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} b^{5} c^{2} - 2 \, \sqrt {b d} a b^{4} c d + \sqrt {b d} a^{2} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a c^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 441, normalized size = 3.56 \[ \frac {\sqrt {b x +a}\, \left (3 a^{2} d^{3} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-2 a b c \,d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-b^{2} c^{2} d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 a^{2} c \,d^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-2 a b \,c^{2} d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-b^{2} c^{3} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,d^{2} x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c d x -2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c d +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2}\right )}{2 \left (a d -b c \right ) \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, a \,c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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