Optimal. Leaf size=148 \[ \frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{7/2} \sqrt {d}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 b^3}+\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}{12 b^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b} \]
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Rubi [A] time = 0.07, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {50, 63, 217, 206} \begin {gather*} \frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 b^3}+\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}{12 b^2}+\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{7/2} \sqrt {d}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}} \, dx &=\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}+\frac {(5 (b c-a d)) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{6 b}\\ &=\frac {5 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}+\frac {\left (5 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{8 b^2}\\ &=\frac {5 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^3}+\frac {5 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}+\frac {\left (5 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^3}\\ &=\frac {5 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^3}+\frac {5 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}+\frac {\left (5 (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 b^4}\\ &=\frac {5 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^3}+\frac {5 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}+\frac {\left (5 (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 b^4}\\ &=\frac {5 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^3}+\frac {5 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}+\frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{7/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 139, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c+d x} \left (\sqrt {a+b x} \left (15 a^2 d^2-10 a b d (4 c+d x)+b^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )+\frac {15 (b c-a d)^{5/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{24 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 160, normalized size = 1.08 \begin {gather*} \frac {5 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{7/2} \sqrt {d}}+\frac {(b c-a d)^3 \left (\frac {33 b^2 \sqrt {a+b x}}{\sqrt {c+d x}}+\frac {15 d^2 (a+b x)^{5/2}}{(c+d x)^{5/2}}-\frac {40 b d (a+b x)^{3/2}}{(c+d x)^{3/2}}\right )}{24 b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 412, normalized size = 2.78 \begin {gather*} \left [-\frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {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 x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{3} d^{3} x^{2} + 33 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3} + 2 \, {\left (13 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{4} d}, -\frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (8 \, b^{3} d^{3} x^{2} + 33 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3} + 2 \, {\left (13 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{4} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.35, size = 446, normalized size = 3.01 \begin {gather*} -\frac {\frac {24 \, {\left (\frac {{\left (b^{2} c - a b d\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 )}{\sqrt {b d}} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}\right )} c^{2} {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} 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 )}{\sqrt {b d} b d^{2}}\right )} d^{2} {\left | b \right |}}{b^{2}} - \frac {12 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b 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 )}{\sqrt {b d} d}\right )} c d {\left | b \right |}}{b^{3}}}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 465, normalized size = 3.14 \begin {gather*} -\frac {5 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{16 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, b^{3}}+\frac {15 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c \,d^{2} \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{16 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, b^{2}}-\frac {15 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,c^{2} d \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{16 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, b}+\frac {5 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, c^{3} \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{16 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}+\frac {5 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} d^{2}}{8 b^{3}}-\frac {5 \sqrt {d x +c}\, \sqrt {b x +a}\, a c d}{4 b^{2}}+\frac {5 \sqrt {d x +c}\, \sqrt {b x +a}\, c^{2}}{8 b}-\frac {5 \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}\, a d}{12 b^{2}}+\frac {5 \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}\, c}{12 b}+\frac {\left (d x +c \right )^{\frac {5}{2}} \sqrt {b x +a}}{3 b} \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 )}^{5/2}}{\sqrt {a+b\,x}} \,d x \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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