Optimal. Leaf size=336 \[ -\frac {A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac {-a^3 d^3 D+a^2 b C d^3+a b^2 d \left (-3 B d^2-6 c^2 D+4 c C d\right )-\left (b^3 \left (-5 A d^3+2 B c d^2-2 c^3 D\right )\right )}{b^2 d^2 \sqrt {c+d x} (b c-a d)^3}+\frac {3 a^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3-\left (b^3 \left (5 A d^3-2 B c d^2-2 c^3 D+2 c^2 C d\right )\right )}{3 b^3 d^2 (c+d x)^{3/2} (b c-a d)^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (a^3 (-d) D-a^2 b (C d-6 c D)-a b^2 (4 c C-3 B d)+b^3 (2 B c-5 A d)\right )}{b^{3/2} (b c-a d)^{7/2}} \]
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Rubi [A] time = 0.79, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1621, 897, 1261, 208} \[ -\frac {a^2 b C d^3-a^3 d^3 D+a b^2 d \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (-\left (-5 A d^3+2 B c d^2-2 c^3 D\right )\right )}{b^2 d^2 \sqrt {c+d x} (b c-a d)^3}+\frac {-3 a^2 b C d^3+3 a^3 d^3 D+3 a b^2 B d^3+b^3 \left (-\left (5 A d^3-2 B c d^2+2 c^2 C d-2 c^3 D\right )\right )}{3 b^3 d^2 (c+d x)^{3/2} (b c-a d)^2}-\frac {A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (-a^2 b (C d-6 c D)+a^3 (-d) D-a b^2 (4 c C-3 B d)+b^3 (2 B c-5 A d)\right )}{b^{3/2} (b c-a d)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 897
Rule 1261
Rule 1621
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{5/2}} \, dx &=-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}+\frac {\int \frac {-\frac {b^3 (2 B c-5 A d)-a b^2 (2 c C-3 B d)+3 a^3 d D-a^2 b (3 C d-2 c D)}{2 b^3}-\frac {(b c-a d) (b C-a D) x}{b^2}-\left (c-\frac {a d}{b}\right ) D x^2}{(a+b x) (c+d x)^{5/2}} \, dx}{-b c+a d}\\ &=-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {2 \operatorname {Subst}\left (\int \frac {\frac {-c^2 \left (c-\frac {a d}{b}\right ) D+\frac {c d (b c-a d) (b C-a D)}{b^2}-\frac {d^2 \left (b^3 (2 B c-5 A d)-a b^2 (2 c C-3 B d)+3 a^3 d D-a^2 b (3 C d-2 c D)\right )}{2 b^3}}{d^2}-\frac {\left (-2 c \left (c-\frac {a d}{b}\right ) D+\frac {d (b c-a d) (b C-a D)}{b^2}\right ) x^2}{d^2}-\frac {\left (c-\frac {a d}{b}\right ) D x^4}{d^2}}{x^4 \left (\frac {-b c+a d}{d}+\frac {b x^2}{d}\right )} \, dx,x,\sqrt {c+d x}\right )}{d (b c-a d)}\\ &=-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {2 \operatorname {Subst}\left (\int \left (\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+5 A d^3-2 c^3 D\right )}{2 b^3 d (b c-a d) x^4}+\frac {-a^2 b C d^3+a^3 d^3 D-a b^2 d \left (4 c C d-3 B d^2-6 c^2 D\right )+b^3 \left (2 B c d^2-5 A d^3-2 c^3 D\right )}{2 b^2 d (b c-a d)^2 x^2}+\frac {d \left (b^3 (2 B c-5 A d)-a b^2 (4 c C-3 B d)-a^3 d D-a^2 b (C d-6 c D)\right )}{2 b (b c-a d)^2 \left (b c-a d-b x^2\right )}\right ) \, dx,x,\sqrt {c+d x}\right )}{d (b c-a d)}\\ &=\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+5 A d^3-2 c^3 D\right )}{3 b^3 d^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {a^2 b C d^3-a^3 d^3 D+a b^2 d \left (4 c C d-3 B d^2-6 c^2 D\right )-b^3 \left (2 B c d^2-5 A d^3-2 c^3 D\right )}{b^2 d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\left (b^3 (2 B c-5 A d)-a b^2 (4 c C-3 B d)-a^3 d D-a^2 b (C d-6 c D)\right ) \operatorname {Subst}\left (\int \frac {1}{b c-a d-b x^2} \, dx,x,\sqrt {c+d x}\right )}{b (b c-a d)^3}\\ &=\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (2 c^2 C d-2 B c d^2+5 A d^3-2 c^3 D\right )}{3 b^3 d^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac {a^2 b C d^3-a^3 d^3 D+a b^2 d \left (4 c C d-3 B d^2-6 c^2 D\right )-b^3 \left (2 B c d^2-5 A d^3-2 c^3 D\right )}{b^2 d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\left (b^3 (2 B c-5 A d)-a b^2 (4 c C-3 B d)-a^3 d D-a^2 b (C d-6 c D)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b c-a d)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 334, normalized size = 0.99 \[ \frac {\sqrt {c+d x} \left (a \left (a^2 D-a b C+b^2 B\right )-A b^3\right )}{b (a+b x) (b c-a d)^3}+\frac {d \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b c-a d)^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (a^3 (-d) D+3 a^2 b c D+a b^2 (B d-2 c C)+b^3 (B c-2 A d)\right )}{b^{3/2} (b c-a d)^{7/2}}+\frac {2 \left (-A d^3+B c d^2+c^3 D-c^2 C d\right )}{3 d^2 (c+d x)^{3/2} (b c-a d)^2}+\frac {b \left (4 A d^3-2 B c d^2+2 c^3 D\right )-2 a d \left (B d^2+3 c^2 D-2 c C d\right )}{d^2 \sqrt {c+d x} (a d-b c)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 2444, normalized size = 7.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.42, size = 439, normalized size = 1.31 \[ \frac {{\left (6 \, D a^{2} b c - 4 \, C a b^{2} c + 2 \, B b^{3} c - D a^{3} d - C a^{2} b d + 3 \, B a b^{2} d - 5 \, A b^{3} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt {-b^{2} c + a b d}} + \frac {\sqrt {d x + c} D a^{3} d - \sqrt {d x + c} C a^{2} b d + \sqrt {d x + c} B a b^{2} d - \sqrt {d x + c} A b^{3} d}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}} - \frac {2 \, {\left (3 \, {\left (d x + c\right )} D b c^{3} - D b c^{4} - 9 \, {\left (d x + c\right )} D a c^{2} d + D a c^{3} d + C b c^{3} d + 6 \, {\left (d x + c\right )} C a c d^{2} - 3 \, {\left (d x + c\right )} B b c d^{2} - C a c^{2} d^{2} - B b c^{2} d^{2} - 3 \, {\left (d x + c\right )} B a d^{3} + 6 \, {\left (d x + c\right )} A b d^{3} + B a c d^{3} + A b c d^{3} - A a d^{4}\right )}}{3 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 730, normalized size = 2.17 \[ \frac {5 A \,b^{2} d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}}-\frac {3 B a b d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}}-\frac {2 B \,b^{2} c \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}}+\frac {C \,a^{2} d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}}+\frac {4 C a b c \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}}+\frac {D a^{3} d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}\, b}-\frac {6 D a^{2} c \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{3} \sqrt {\left (a d -b c \right ) b}}+\frac {\sqrt {d x +c}\, A \,b^{2} d}{\left (a d -b c \right )^{3} \left (b d x +a d \right )}-\frac {\sqrt {d x +c}\, B a b d}{\left (a d -b c \right )^{3} \left (b d x +a d \right )}+\frac {\sqrt {d x +c}\, C \,a^{2} d}{\left (a d -b c \right )^{3} \left (b d x +a d \right )}-\frac {\sqrt {d x +c}\, D a^{3} d}{\left (a d -b c \right )^{3} \left (b d x +a d \right ) b}+\frac {4 A b d}{\left (a d -b c \right )^{3} \sqrt {d x +c}}-\frac {2 B a d}{\left (a d -b c \right )^{3} \sqrt {d x +c}}-\frac {2 B b c}{\left (a d -b c \right )^{3} \sqrt {d x +c}}+\frac {4 C a c}{\left (a d -b c \right )^{3} \sqrt {d x +c}}-\frac {6 D a \,c^{2}}{\left (a d -b c \right )^{3} \sqrt {d x +c}\, d}+\frac {2 D b \,c^{3}}{\left (a d -b c \right )^{3} \sqrt {d x +c}\, d^{2}}-\frac {2 A d}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 B c}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 C \,c^{2}}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}} d}+\frac {2 D c^{3}}{3 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}} d^{2}} \]
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 \[ \int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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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