Optimal. Leaf size=192 \[ -\frac {5 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{7/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}{64 b d^3}-\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2}{96 b d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}{24 b d}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b} \]
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Rubi [A] time = 0.10, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {50, 63, 217, 206} \[ -\frac {5 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{7/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}{64 b d^3}-\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2}{96 b d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}{24 b d}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int (a+b x)^{5/2} \sqrt {c+d x} \, dx &=\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}+\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{8 b}\\ &=\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b d}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}-\frac {\left (5 (b c-a d)^2\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{48 b d}\\ &=-\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^2}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b d}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}+\frac {\left (5 (b c-a d)^3\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b d^2}\\ &=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b d^3}-\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^2}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b d}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}-\frac {\left (5 (b c-a d)^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b d^3}\\ &=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b d^3}-\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^2}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b d}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}-\frac {\left (5 (b c-a d)^4\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 )}{64 b^2 d^3}\\ &=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b d^3}-\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^2}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b d}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}-\frac {\left (5 (b c-a d)^4\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 )}{64 b^2 d^3}\\ &=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b d^3}-\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^2}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 b d}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}-\frac {5 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 190, normalized size = 0.99 \[ \frac {b \sqrt {d} \sqrt {a+b x} (c+d x) \left (15 a^3 d^3+a^2 b d^2 (73 c+118 d x)+a b^2 d \left (-55 c^2+36 c d x+136 d^2 x^2\right )+b^3 \left (15 c^3-10 c^2 d x+8 c d^2 x^2+48 d^3 x^3\right )\right )-15 (b c-a d)^{9/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{192 b^2 d^{7/2} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 540, normalized size = 2.81 \[ \left [\frac {15 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 55 \, a b^{3} c^{2} d^{2} + 73 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \, {\left (b^{4} c d^{3} + 17 \, a b^{3} d^{4}\right )} x^{2} - 2 \, {\left (5 \, b^{4} c^{2} d^{2} - 18 \, a b^{3} c d^{3} - 59 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{2} d^{4}}, \frac {15 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 55 \, a b^{3} c^{2} d^{2} + 73 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \, {\left (b^{4} c d^{3} + 17 \, a b^{3} d^{4}\right )} x^{2} - 2 \, {\left (5 \, b^{4} c^{2} d^{2} - 18 \, a b^{3} c d^{3} - 59 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{2} d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.02, size = 726, normalized size = 3.78 \[ \frac {24 \, {\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 )} a {\left | b \right |} - \frac {192 \, {\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^{3} {\left | b \right |}}{b^{2}} + {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\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^{2} d^{3}}\right )} b {\left | b \right |} + \frac {144 \, {\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^{2} {\left | b \right |}}{b^{2}}}{192 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 645, normalized size = 3.36 \[ -\frac {5 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} 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 )}{128 \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 )}\, a^{3} 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 )}{32 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}-\frac {15 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} 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 )}{64 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, d}+\frac {5 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,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 )}{32 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, d^{2}}-\frac {5 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{4} \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 )}{128 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, d^{3}}+\frac {5 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3}}{64 b}-\frac {15 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} c}{64 d}+\frac {15 \sqrt {d x +c}\, \sqrt {b x +a}\, a b \,c^{2}}{64 d^{2}}-\frac {5 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c^{3}}{64 d^{3}}+\frac {5 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} a^{2}}{32 d}-\frac {5 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} a b c}{16 d^{2}}+\frac {5 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} b^{2} c^{2}}{32 d^{3}}+\frac {5 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} a}{24 d}-\frac {5 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} b c}{24 d^{2}}+\frac {\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}}}{4 d} \]
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 \[ \int {\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x} \,d x \]
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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