Optimal. Leaf size=138 \[ \frac {15 \sqrt {d} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2}}+\frac {15 d \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 b^3}+\frac {5 d \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2}-\frac {2 (c+d x)^{5/2}}{b \sqrt {a+b x}} \]
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Rubi [A] time = 0.07, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {5 d \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2}+\frac {15 d \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 b^3}+\frac {15 \sqrt {d} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2}}-\frac {2 (c+d x)^{5/2}}{b \sqrt {a+b x}} \]
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)^{5/2}}{(a+b x)^{3/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {(5 d) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{b}\\ &=\frac {5 d \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2}-\frac {2 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {(15 d (b c-a d)) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{4 b^2}\\ &=\frac {15 d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {5 d \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2}-\frac {2 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {\left (15 d (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^3}\\ &=\frac {15 d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {5 d \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2}-\frac {2 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {\left (15 d (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b^4}\\ &=\frac {15 d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {5 d \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2}-\frac {2 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {\left (15 d (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b^4}\\ &=\frac {15 d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {5 d \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2}-\frac {2 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {15 \sqrt {d} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 71, normalized size = 0.51 \[ -\frac {2 (c+d x)^{5/2} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 439, normalized size = 3.18 \[ \left [\frac {15 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 25 \, a b c d - 15 \, a^{2} d^{2} + {\left (9 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (b^{4} x + a b^{3}\right )}}, -\frac {15 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 25 \, a b c d - 15 \, a^{2} d^{2} + {\left (9 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (b^{4} x + a b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.77, size = 287, normalized size = 2.08 \[ \frac {1}{4} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{5}} + \frac {9 \, {\left (b^{10} c d^{3} {\left | b \right |} - a b^{9} d^{4} {\left | b \right |}\right )}}{b^{14} d^{2}}\right )} - \frac {15 \, {\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a b c d {\left | b \right |} + \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, b^{5}} - \frac {4 \, {\left (\sqrt {b d} b^{3} c^{3} {\left | b \right |} - 3 \, \sqrt {b d} a b^{2} c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} a^{2} b c d^{2} {\left | b \right |} - \sqrt {b d} a^{3} d^{3} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{\frac {5}{2}}}{\left (b x +a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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