Optimal. Leaf size=174 \[ -\frac {35 \sqrt {b} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{9/2}}+\frac {35 b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 d^4}-\frac {35 b (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{12 d^3}+\frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}} \]
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Rubi [A] time = 0.09, antiderivative size = 174, 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} \begin {gather*} \frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}-\frac {35 b (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{12 d^3}+\frac {35 b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 d^4}-\frac {35 \sqrt {b} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{9/2}}-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d 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 {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx &=-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}+\frac {(7 b) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}+\frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}-\frac {(35 b (b c-a d)) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{6 d^2}\\ &=-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}-\frac {35 b (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}+\frac {\left (35 b (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 d^3}\\ &=-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}+\frac {35 b (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {35 b (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}-\frac {\left (35 b (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}+\frac {35 b (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {35 b (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}-\frac {\left (35 (b c-a d)^3\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 )}{8 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}+\frac {35 b (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {35 b (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}-\frac {\left (35 (b c-a d)^3\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 )}{8 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}+\frac {35 b (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {35 b (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}-\frac {35 \sqrt {b} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 73, normalized size = 0.42 \begin {gather*} \frac {2 (a+b x)^{9/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {d (a+b x)}{a d-b c}\right )}{9 b (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 173, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {a+b x} (a d-b c)^3 \left (\frac {280 b^2 d (a+b x)}{c+d x}+\frac {48 d^3 (a+b x)^3}{(c+d x)^3}-\frac {231 b d^2 (a+b x)^2}{(c+d x)^2}-105 b^3\right )}{24 d^4 \sqrt {c+d x} \left (\frac {d (a+b x)}{c+d x}-b\right )^3}-\frac {35 \sqrt {b} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.73, size = 603, normalized size = 3.47 \begin {gather*} \left [-\frac {105 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} + {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\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 (8 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} - 280 \, a b^{2} c^{2} d + 231 \, a^{2} b c d^{2} - 48 \, a^{3} d^{3} - 2 \, {\left (7 \, b^{3} c d^{2} - 19 \, a b^{2} d^{3}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d - 98 \, a b^{2} c d^{2} + 87 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (d^{5} x + c d^{4}\right )}}, \frac {105 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} + {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\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 (8 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} - 280 \, a b^{2} c^{2} d + 231 \, a^{2} b c d^{2} - 48 \, a^{3} d^{3} - 2 \, {\left (7 \, b^{3} c d^{2} - 19 \, a b^{2} d^{3}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d - 98 \, a b^{2} c d^{2} + 87 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (d^{5} x + c d^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.38, size = 279, normalized size = 1.60 \begin {gather*} \frac {{\left ({\left (b x + a\right )} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} b^{2}}{d {\left | b \right |}} - \frac {7 \, {\left (b^{3} c d^{5} - a b^{2} d^{6}\right )}}{d^{7} {\left | b \right |}}\right )} + \frac {35 \, {\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )}}{d^{7} {\left | b \right |}}\right )} + \frac {105 \, {\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{d^{7} {\left | b \right |}}\right )} \sqrt {b x + a}}{24 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {35 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\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 )}{8 \, \sqrt {b d} d^{4} {\left | b \right |}} \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 (b x +a \right )^{\frac {7}{2}}}{\left (d x +c \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 (a+b\,x\right )}^{7/2}}{{\left (c+d\,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 (a + b x\right )^{\frac {7}{2}}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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