Optimal. Leaf size=98 \[ \frac {3 \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}+\frac {3 d \sqrt {a+b x} \sqrt {c+d x}}{b^2}-\frac {2 (c+d x)^{3/2}}{b \sqrt {a+b x}} \]
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Rubi [A] time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {47, 50, 63, 217, 206} \begin {gather*} \frac {3 d \sqrt {a+b x} \sqrt {c+d x}}{b^2}+\frac {3 \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}-\frac {2 (c+d x)^{3/2}}{b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx &=-\frac {2 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {(3 d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{b}\\ &=\frac {3 d \sqrt {a+b x} \sqrt {c+d x}}{b^2}-\frac {2 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {(3 d (b c-a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^2}\\ &=\frac {3 d \sqrt {a+b x} \sqrt {c+d x}}{b^2}-\frac {2 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {(3 d (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^3}\\ &=\frac {3 d \sqrt {a+b x} \sqrt {c+d x}}{b^2}-\frac {2 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {(3 d (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^3}\\ &=\frac {3 d \sqrt {a+b x} \sqrt {c+d x}}{b^2}-\frac {2 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {3 \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 71, normalized size = 0.72 \begin {gather*} -\frac {2 (c+d x)^{3/2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 159, normalized size = 1.62 \begin {gather*} \frac {\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}} \left (-3 a d^2 \sqrt {c+d x}-b d (c+d x)^{3/2}+3 b c d \sqrt {c+d x}\right )}{b^2 (-a d-b (c+d x)+b c)}-\frac {3 \sqrt {\frac {b}{d}} \left (b c d-a d^2\right ) \log \left (\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 311, normalized size = 3.17 \begin {gather*} \left [-\frac {3 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (b d x - 2 \, b c + 3 \, a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (b^{3} x + a b^{2}\right )}}, -\frac {3 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - 2 \, {\left (b d x - 2 \, b c + 3 \, a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{3} x + a b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.50, size = 204, normalized size = 2.08 \begin {gather*} \frac {\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} d {\left | b \right |}}{b^{4}} - \frac {3 \, {\left (\sqrt {b d} b c {\left | b \right |} - \sqrt {b d} a d {\left | b \right |}\right )} \log \left ({\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 )}{2 \, b^{4}} - \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 )} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d x +c \right )^{\frac {3}{2}}}{\left (b x +a \right )^{\frac {3}{2}}}\, dx \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{3/2}}{{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\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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