Optimal. Leaf size=154 \[ \frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 b d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{12 b d}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b} \]
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Rubi [A] time = 0.07, antiderivative size = 154, 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 {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 b d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{12 b d}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{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 (a+b x)^{3/2} \sqrt {c+d x} \, dx &=\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}+\frac {(b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{6 b}\\ &=\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}-\frac {(b c-a d)^2 \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 b d}\\ &=-\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b d^2}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}+\frac {(b c-a d)^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b d^2}\\ &=-\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b d^2}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}+\frac {(b c-a d)^3 \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^2 d^2}\\ &=-\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b d^2}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}+\frac {(b c-a d)^3 \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^2 d^2}\\ &=-\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b d^2}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}+\frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 151, normalized size = 0.98 \begin {gather*} \frac {b \sqrt {d} \sqrt {a+b x} (c+d x) \left (3 a^2 d^2+2 a b d (4 c+7 d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )+3 (b c-a d)^{7/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{24 b^2 d^{5/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 154, normalized size = 1.00 \begin {gather*} \frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{3/2} d^{5/2}}-\frac {\sqrt {c+d x} (b c-a d)^3 \left (\frac {3 b^2 (c+d x)^2}{(a+b x)^2}-\frac {8 b d (c+d x)}{a+b x}-3 d^2\right )}{24 b d^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.55, size = 410, normalized size = 2.66 \begin {gather*} \left [-\frac {3 \, {\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} - 3 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} + 2 \, {\left (b^{3} c d^{2} + 7 \, a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{2} d^{3}}, -\frac {3 \, {\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} - 3 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} + 2 \, {\left (b^{3} c d^{2} + 7 \, a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{2} d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.20, size = 438, normalized size = 2.84 \begin {gather*} \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 )} {\left | b \right |} - \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 )} a^{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 )} a {\left | b \right |}}{b^{2}}}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 460, normalized size = 2.99 \begin {gather*} -\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} 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 {3 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c \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 {3 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{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}\, d}+\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} 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}\, d^{2}}+\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, a^{2}}{8 b}-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, a c}{4 d}+\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, b \,c^{2}}{8 d^{2}}+\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} a}{4 d}-\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} b c}{4 d^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}{3 d} \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 {\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,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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