Optimal. Leaf size=170 \[ \frac {35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}}-\frac {35 b^2 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}-\frac {35 b^2 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 d^4}+\frac {35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{7/2}}{(c+d x)^{5/2}} \, dx &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}+\frac {(7 b) \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}+\frac {\left (35 b^2\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}-\frac {\left (35 b^2 (b c-a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 d^3}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {\left (35 b^2 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {\left (35 b (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 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {\left (35 b (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 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 73, normalized size = 0.43 \[ \frac {2 (a+b x)^{9/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} \, _2F_1\left (\frac {5}{2},\frac {9}{2};\frac {11}{2};\frac {d (a+b x)}{a d-b c}\right )}{9 b (c+d x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 657, normalized size = 3.86 \[ \left [\frac {105 \, {\left (b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} + {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (6 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 175 \, a b^{2} c^{2} d - 56 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3} - 3 \, {\left (7 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2} - 2 \, {\left (70 \, b^{3} c^{2} d - 119 \, a b^{2} c d^{2} + 40 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}, -\frac {105 \, {\left (b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} + {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - 2 \, {\left (6 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 175 \, a b^{2} c^{2} d - 56 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3} - 3 \, {\left (7 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2} - 2 \, {\left (70 \, b^{3} c^{2} d - 119 \, a b^{2} c d^{2} + 40 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.14, size = 380, normalized size = 2.24 \[ \frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{6} c d^{6} - a b^{5} d^{7}\right )} {\left (b x + a\right )}}{b^{2} c d^{7} {\left | b \right |} - a b d^{8} {\left | b \right |}} - \frac {7 \, {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )}}{b^{2} c d^{7} {\left | b \right |} - a b d^{8} {\left | b \right |}}\right )} - \frac {140 \, {\left (b^{8} c^{3} d^{4} - 3 \, a b^{7} c^{2} d^{5} + 3 \, a^{2} b^{6} c d^{6} - a^{3} b^{5} d^{7}\right )}}{b^{2} c d^{7} {\left | b \right |} - a b d^{8} {\left | b \right |}}\right )} {\left (b x + a\right )} - \frac {105 \, {\left (b^{9} c^{4} d^{3} - 4 \, a b^{8} c^{3} d^{4} + 6 \, a^{2} b^{7} c^{2} d^{5} - 4 \, a^{3} b^{6} c d^{6} + a^{4} b^{5} d^{7}\right )}}{b^{2} c d^{7} {\left | b \right |} - a b d^{8} {\left | b \right |}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {35 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} d^{4} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {7}{2}}}{\left (d x +c \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{7/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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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