Optimal. Leaf size=119 \[ -\frac {15 d^2 \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{7/2}}-\frac {5 d (c+d x)^{3/2}}{4 b^2 (a+b x)}-\frac {(c+d x)^{5/2}}{2 b (a+b x)^2}+\frac {15 d^2 \sqrt {c+d x}}{4 b^3} \]
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Rubi [A] time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {47, 50, 63, 208} \begin {gather*} -\frac {15 d^2 \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{7/2}}-\frac {5 d (c+d x)^{3/2}}{4 b^2 (a+b x)}-\frac {(c+d x)^{5/2}}{2 b (a+b x)^2}+\frac {15 d^2 \sqrt {c+d x}}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{(a+b x)^3} \, dx &=-\frac {(c+d x)^{5/2}}{2 b (a+b x)^2}+\frac {(5 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac {5 d (c+d x)^{3/2}}{4 b^2 (a+b x)}-\frac {(c+d x)^{5/2}}{2 b (a+b x)^2}+\frac {\left (15 d^2\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{8 b^2}\\ &=\frac {15 d^2 \sqrt {c+d x}}{4 b^3}-\frac {5 d (c+d x)^{3/2}}{4 b^2 (a+b x)}-\frac {(c+d x)^{5/2}}{2 b (a+b x)^2}+\frac {\left (15 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{8 b^3}\\ &=\frac {15 d^2 \sqrt {c+d x}}{4 b^3}-\frac {5 d (c+d x)^{3/2}}{4 b^2 (a+b x)}-\frac {(c+d x)^{5/2}}{2 b (a+b x)^2}+\frac {(15 d (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b^3}\\ &=\frac {15 d^2 \sqrt {c+d x}}{4 b^3}-\frac {5 d (c+d x)^{3/2}}{4 b^2 (a+b x)}-\frac {(c+d x)^{5/2}}{2 b (a+b x)^2}-\frac {15 d^2 \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.44 \begin {gather*} \frac {2 d^2 (c+d x)^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};-\frac {b (c+d x)}{a d-b c}\right )}{7 (a d-b c)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 155, normalized size = 1.30 \begin {gather*} \frac {d^2 \sqrt {c+d x} \left (15 a^2 d^2+25 a b d (c+d x)-30 a b c d+15 b^2 c^2+8 b^2 (c+d x)^2-25 b^2 c (c+d x)\right )}{4 b^3 (a d+b (c+d x)-b c)^2}+\frac {15 d^2 \sqrt {a d-b c} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{4 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 344, normalized size = 2.89 \begin {gather*} \left [\frac {15 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (8 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} - 5 \, a b c d + 15 \, a^{2} d^{2} - {\left (9 \, b^{2} c d - 25 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {15 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (8 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} - 5 \, a b c d + 15 \, a^{2} d^{2} - {\left (9 \, b^{2} c d - 25 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 171, normalized size = 1.44 \begin {gather*} \frac {2 \, \sqrt {d x + c} d^{2}}{b^{3}} + \frac {15 \, {\left (b c d^{2} - a d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, \sqrt {-b^{2} c + a b d} b^{3}} - \frac {9 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{2} - 7 \, \sqrt {d x + c} b^{2} c^{2} d^{2} - 9 \, {\left (d x + c\right )}^{\frac {3}{2}} a b d^{3} + 14 \, \sqrt {d x + c} a b c d^{3} - 7 \, \sqrt {d x + c} a^{2} d^{4}}{4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 238, normalized size = 2.00 \begin {gather*} \frac {7 \sqrt {d x +c}\, a^{2} d^{4}}{4 \left (b d x +a d \right )^{2} b^{3}}-\frac {7 \sqrt {d x +c}\, a c \,d^{3}}{2 \left (b d x +a d \right )^{2} b^{2}}+\frac {7 \sqrt {d x +c}\, c^{2} d^{2}}{4 \left (b d x +a d \right )^{2} b}+\frac {9 \left (d x +c \right )^{\frac {3}{2}} a \,d^{3}}{4 \left (b d x +a d \right )^{2} b^{2}}-\frac {15 a \,d^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{4 \sqrt {\left (a d -b c \right ) b}\, b^{3}}-\frac {9 \left (d x +c \right )^{\frac {3}{2}} c \,d^{2}}{4 \left (b d x +a d \right )^{2} b}+\frac {15 c \,d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{4 \sqrt {\left (a d -b c \right ) b}\, b^{2}}+\frac {2 \sqrt {d x +c}\, d^{2}}{b^{3}} \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 [B] time = 0.16, size = 199, normalized size = 1.67 \begin {gather*} \frac {2\,d^2\,\sqrt {c+d\,x}}{b^3}-\frac {\left (\frac {9\,b^2\,c\,d^2}{4}-\frac {9\,a\,b\,d^3}{4}\right )\,{\left (c+d\,x\right )}^{3/2}-\sqrt {c+d\,x}\,\left (\frac {7\,a^2\,d^4}{4}-\frac {7\,a\,b\,c\,d^3}{2}+\frac {7\,b^2\,c^2\,d^2}{4}\right )}{b^5\,{\left (c+d\,x\right )}^2-\left (2\,b^5\,c-2\,a\,b^4\,d\right )\,\left (c+d\,x\right )+b^5\,c^2+a^2\,b^3\,d^2-2\,a\,b^4\,c\,d}-\frac {15\,d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,d^2\,\sqrt {a\,d-b\,c}\,\sqrt {c+d\,x}}{a\,d^3-b\,c\,d^2}\right )\,\sqrt {a\,d-b\,c}}{4\,b^{7/2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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