Optimal. Leaf size=193 \[ -\frac {2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}+\frac {2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^3 \sqrt {c+d x} (b c-a d)}+\frac {2 \sqrt {c+d x} (-a d D-b c D+b C d)}{b^2 d^3}+\frac {2 D (c+d x)^{3/2}}{3 b d^3}-\frac {2 c D \sqrt {c+d x}}{b d^3} \]
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Rubi [A] time = 0.24, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1619, 43, 63, 208} \[ -\frac {2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}+\frac {2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^3 \sqrt {c+d x} (b c-a d)}+\frac {2 \sqrt {c+d x} (-a d D-b c D+b C d)}{b^2 d^3}+\frac {2 D (c+d x)^{3/2}}{3 b d^3}-\frac {2 c D \sqrt {c+d x}}{b d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 63
Rule 208
Rule 1619
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{3/2}} \, dx &=\int \left (\frac {c^2 C d-B c d^2+A d^3-c^3 D}{d^2 (-b c+a d) (c+d x)^{3/2}}+\frac {b C d-b c D-a d D}{b^2 d^2 \sqrt {c+d x}}+\frac {D x}{b d \sqrt {c+d x}}+\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{b^2 (b c-a d) (a+b x) \sqrt {c+d x}}\right ) \, dx\\ &=\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^3 (b c-a d) \sqrt {c+d x}}+\frac {2 (b C d-b c D-a d D) \sqrt {c+d x}}{b^2 d^3}+\frac {D \int \frac {x}{\sqrt {c+d x}} \, dx}{b d}+\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^2 (b c-a d)}\\ &=\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^3 (b c-a d) \sqrt {c+d x}}+\frac {2 (b C d-b c D-a d D) \sqrt {c+d x}}{b^2 d^3}+\frac {D \int \left (-\frac {c}{d \sqrt {c+d x}}+\frac {\sqrt {c+d x}}{d}\right ) \, dx}{b d}+\frac {\left (2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^2 d (b c-a d)}\\ &=\frac {2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^3 (b c-a d) \sqrt {c+d x}}-\frac {2 c D \sqrt {c+d x}}{b d^3}+\frac {2 (b C d-b c D-a d D) \sqrt {c+d x}}{b^2 d^3}+\frac {2 D (c+d x)^{3/2}}{3 b d^3}-\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 174, normalized size = 0.90 \[ 2 \left (-\frac {\left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{d^3 \sqrt {c+d x} (b c-a d)}+\frac {\sqrt {c+d x} (-a d D-2 b c D+b C d)}{b^2 d^3}+\frac {D (c+d x)^{3/2}}{3 b d^3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 866, normalized size = 4.49 \[ \left [-\frac {3 \, {\left ({\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} d^{4} x + {\left (D a^{3} c - {\left (C a^{2} b - B a b^{2} + A b^{3}\right )} c\right )} d^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (8 \, D b^{4} c^{4} + 3 \, A a b^{3} d^{4} + 3 \, {\left (D a^{3} b c - {\left (C a^{2} b^{2} + B a b^{3} + A b^{4}\right )} c\right )} d^{3} - {\left (D a^{2} b^{2} c^{2} - 3 \, {\left (3 \, C a b^{3} + B b^{4}\right )} c^{2}\right )} d^{2} - {\left (D b^{4} c^{2} d^{2} - 2 \, D a b^{3} c d^{3} + D a^{2} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (5 \, D a b^{3} c^{3} + 3 \, C b^{4} c^{3}\right )} d + {\left (4 \, D b^{4} c^{3} d + 3 \, {\left (D a^{3} b - C a^{2} b^{2}\right )} d^{4} - 2 \, {\left (D a^{2} b^{2} c - 3 \, C a b^{3} c\right )} d^{3} - {\left (5 \, D a b^{3} c^{2} + 3 \, C b^{4} c^{2}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{3 \, {\left (b^{5} c^{3} d^{3} - 2 \, a b^{4} c^{2} d^{4} + a^{2} b^{3} c d^{5} + {\left (b^{5} c^{2} d^{4} - 2 \, a b^{4} c d^{5} + a^{2} b^{3} d^{6}\right )} x\right )}}, -\frac {2 \, {\left (3 \, {\left ({\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} d^{4} x + {\left (D a^{3} c - {\left (C a^{2} b - B a b^{2} + A b^{3}\right )} c\right )} d^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (8 \, D b^{4} c^{4} + 3 \, A a b^{3} d^{4} + 3 \, {\left (D a^{3} b c - {\left (C a^{2} b^{2} + B a b^{3} + A b^{4}\right )} c\right )} d^{3} - {\left (D a^{2} b^{2} c^{2} - 3 \, {\left (3 \, C a b^{3} + B b^{4}\right )} c^{2}\right )} d^{2} - {\left (D b^{4} c^{2} d^{2} - 2 \, D a b^{3} c d^{3} + D a^{2} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (5 \, D a b^{3} c^{3} + 3 \, C b^{4} c^{3}\right )} d + {\left (4 \, D b^{4} c^{3} d + 3 \, {\left (D a^{3} b - C a^{2} b^{2}\right )} d^{4} - 2 \, {\left (D a^{2} b^{2} c - 3 \, C a b^{3} c\right )} d^{3} - {\left (5 \, D a b^{3} c^{2} + 3 \, C b^{4} c^{2}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}\right )}}{3 \, {\left (b^{5} c^{3} d^{3} - 2 \, a b^{4} c^{2} d^{4} + a^{2} b^{3} c d^{5} + {\left (b^{5} c^{2} d^{4} - 2 \, a b^{4} c d^{5} + a^{2} b^{3} d^{6}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.33, size = 200, normalized size = 1.04 \[ -\frac {2 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{3} c - a b^{2} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{{\left (b c d^{3} - a d^{4}\right )} \sqrt {d x + c}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} D b^{2} d^{6} - 6 \, \sqrt {d x + c} D b^{2} c d^{6} - 3 \, \sqrt {d x + c} D a b d^{7} + 3 \, \sqrt {d x + c} C b^{2} d^{7}\right )}}{3 \, b^{3} d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 366, normalized size = 1.90 \[ -\frac {2 A b \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}+\frac {2 B a \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}-\frac {2 C \,a^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}\, b}+\frac {2 D a^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}\, b^{2}}-\frac {2 A}{\left (a d -b c \right ) \sqrt {d x +c}}+\frac {2 B c}{\left (a d -b c \right ) \sqrt {d x +c}\, d}-\frac {2 C \,c^{2}}{\left (a d -b c \right ) \sqrt {d x +c}\, d^{2}}+\frac {2 D c^{3}}{\left (a d -b c \right ) \sqrt {d x +c}\, d^{3}}+\frac {2 \sqrt {d x +c}\, C}{b \,d^{2}}-\frac {2 \sqrt {d x +c}\, D a}{b^{2} d^{2}}-\frac {4 \sqrt {d x +c}\, D c}{b \,d^{3}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} D}{3 b \,d^{3}} \]
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 \[ \int \frac {A+B\,x+C\,x^2+x^3\,D}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 85.73, size = 172, normalized size = 0.89 \[ \frac {2 D \left (c + d x\right )^{\frac {3}{2}}}{3 b d^{3}} + \frac {2 \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{3} \sqrt {c + d x} \left (a d - b c\right )} + \frac {\sqrt {c + d x} \left (2 C b d - 2 D a d - 4 D b c\right )}{b^{2} d^{3}} + \frac {2 \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{3} \sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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