Optimal. Leaf size=108 \[ \frac {8 (2 b d-a e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {6 d^2 \sqrt {a+b x}}{\sqrt {d+e x} (b d-a e)}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b} \]
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Rubi [A] time = 0.12, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {949, 80, 63, 217, 206} \[ \frac {8 (2 b d-a e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {6 d^2 \sqrt {a+b x}}{\sqrt {d+e x} (b d-a e)}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 206
Rule 217
Rule 949
Rubi steps
\begin {align*} \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx &=\frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {2 \int \frac {6 d (b d-a e)+4 e (b d-a e) x}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b d-a e}\\ &=\frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {(4 (2 b d-a e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b}\\ &=\frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {(8 (2 b d-a e)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^2}\\ &=\frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {(8 (2 b d-a e)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b^2}\\ &=\frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {8 (2 b d-a e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 134, normalized size = 1.24 \[ \frac {2 \left (\frac {b \sqrt {a+b x} (b d (7 d+4 e x)-4 a e (d+e x))}{b d-a e}+\frac {4 \sqrt {b d-a e} (2 b d-a e) \sqrt {\frac {b (d+e x)}{b d-a e}} \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{\sqrt {e}}\right )}{b^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.03, size = 463, normalized size = 4.29 \[ \left [-\frac {2 \, {\left ({\left (2 \, b^{2} d^{3} - 3 \, a b d^{2} e + a^{2} d e^{2} + {\left (2 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - {\left (7 \, b^{2} d^{2} e - 4 \, a b d e^{2} + 4 \, {\left (b^{2} d e^{2} - a b e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{2} e - a b^{2} d e^{2} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x}, -\frac {2 \, {\left (2 \, {\left (2 \, b^{2} d^{3} - 3 \, a b d^{2} e + a^{2} d e^{2} + {\left (2 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - {\left (7 \, b^{2} d^{2} e - 4 \, a b d e^{2} + 4 \, {\left (b^{2} d e^{2} - a b e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{2} e - a b^{2} d e^{2} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 193, normalized size = 1.79 \[ -\frac {8 \, {\left (2 \, b d - a e\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b} {\left | b \right |}} + \frac {2 \, \sqrt {b x + a} {\left (\frac {4 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} {\left (b x + a\right )}}{b^{3} d {\left | b \right |} e^{2} - a b^{2} {\left | b \right |} e^{3}} + \frac {7 \, b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 4 \, a^{2} b^{2} e^{4}}{b^{3} d {\left | b \right |} e^{2} - a b^{2} {\left | b \right |} e^{3}}\right )}}{\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 438, normalized size = 4.06 \[ -\frac {2 \sqrt {b x +a}\, \left (2 a^{2} e^{3} x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-6 a b d \,e^{2} x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+4 b^{2} d^{2} e x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+2 a^{2} d \,e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-6 a b \,d^{2} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+4 b^{2} d^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,e^{2} x +4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b d e x -4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a d e +7 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b \,d^{2}\right )}{\sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e x +d}\, b} \]
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 \[ \int \frac {15\,d^2+20\,d\,e\,x+8\,e^2\,x^2}{\sqrt {a+b\,x}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt {a + b x} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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