Optimal. Leaf size=230 \[ \frac {7 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{3/2} d^{9/2}}-\frac {7 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4}{128 b d^4}+\frac {7 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^3}{192 b d^3}-\frac {7 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{240 b d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x} (b c-a d)}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b} \]
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Rubi [A] time = 0.16, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {50, 63, 217, 206} \[ \frac {7 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{3/2} d^{9/2}}-\frac {7 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4}{128 b d^4}+\frac {7 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^3}{192 b d^3}-\frac {7 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{240 b d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x} (b c-a d)}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int (a+b x)^{7/2} \sqrt {c+d x} \, dx &=\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {(b c-a d) \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx}{10 b}\\ &=\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}-\frac {\left (7 (b c-a d)^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{80 b d}\\ &=-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {\left (7 (b c-a d)^3\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{96 b d^2}\\ &=\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}-\frac {\left (7 (b c-a d)^4\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b d^3}\\ &=-\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {\left (7 (b c-a d)^5\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b d^4}\\ &=-\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {\left (7 (b c-a d)^5\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 )}{128 b^2 d^4}\\ &=-\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {\left (7 (b c-a d)^5\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 )}{128 b^2 d^4}\\ &=-\frac {7 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b d^4}+\frac {7 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{192 b d^3}-\frac {7 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{240 b d^2}+\frac {(b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b d}+\frac {(a+b x)^{9/2} \sqrt {c+d x}}{5 b}+\frac {7 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{3/2} d^{9/2}}\\ \end {align*}
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Mathematica [A] time = 1.47, size = 194, normalized size = 0.84 \[ \frac {(a+b x)^{9/2} \sqrt {c+d x} \left (\frac {70 (b c-a d)^{9/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{9/2} (a+b x)^{9/2} \sqrt {\frac {b (c+d x)}{b c-a d}}}-\frac {70 (b c-a d)^4}{d^4 (a+b x)^4}+\frac {140 (b c-a d)^3}{3 d^3 (a+b x)^3}-\frac {112 (b c-a d)^2}{3 d^2 (a+b x)^2}+\frac {32 b c-32 a d}{a d+b d x}+256\right )}{1280 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 702, normalized size = 3.05 \[ \left [-\frac {105 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 490 \, a b^{4} c^{3} d^{2} - 896 \, a^{2} b^{3} c^{2} d^{3} + 790 \, a^{3} b^{2} c d^{4} + 105 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 31 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 32 \, a b^{4} c d^{4} - 263 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 161 \, a b^{4} c^{2} d^{3} + 289 \, a^{2} b^{3} c d^{4} + 605 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{2} d^{5}}, -\frac {105 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 490 \, a b^{4} c^{3} d^{2} - 896 \, a^{2} b^{3} c^{2} d^{3} + 790 \, a^{3} b^{2} c d^{4} + 105 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 31 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 32 \, a b^{4} c d^{4} - 263 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 161 \, a b^{4} c^{2} d^{3} + 289 \, a^{2} b^{3} c d^{4} + 605 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{2} d^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.84, size = 1107, normalized size = 4.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 858, normalized size = 3.73 \[ -\frac {7 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{5} 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 )}{256 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, b}+\frac {35 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} 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 )}{256 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}-\frac {35 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} 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 )}{128 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, d}+\frac {35 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} 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 )}{128 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, d^{2}}-\frac {35 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,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 )}{256 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, d^{3}}+\frac {7 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{4} c^{5} \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 )}{256 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, d^{4}}+\frac {7 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{4}}{128 b}-\frac {7 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} c}{32 d}+\frac {21 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b \,c^{2}}{64 d^{2}}-\frac {7 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{2} c^{3}}{32 d^{3}}+\frac {7 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{3} c^{4}}{128 d^{4}}+\frac {7 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} a^{3}}{64 d}-\frac {21 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} a^{2} b c}{64 d^{2}}+\frac {21 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} a \,b^{2} c^{2}}{64 d^{3}}-\frac {7 \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} b^{3} c^{3}}{64 d^{4}}+\frac {7 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} a^{2}}{48 d}-\frac {7 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} a b c}{24 d^{2}}+\frac {7 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} b^{2} c^{2}}{48 d^{3}}+\frac {7 \left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}} a}{40 d}-\frac {7 \left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}} b c}{40 d^{2}}+\frac {\left (b x +a \right )^{\frac {7}{2}} \left (d x +c \right )^{\frac {3}{2}}}{5 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 = -0.00 \[ \int {\left (a+b\,x\right )}^{7/2}\,\sqrt {c+d\,x} \,d x \]
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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