Optimal. Leaf size=298 \[ \frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^4 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {16 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 e^4 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {732, 812, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^3 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^4 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {16 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 e^4 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 732
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {\int \frac {(b+2 c x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx}{e}\\ &=\frac {2 (8 c d-3 b e+2 c e x) \sqrt {b x+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {2 \int \frac {\frac {1}{2} b (8 c d-3 b e)+4 c (2 c d-b e) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 e^3}\\ &=\frac {2 (8 c d-3 b e+2 c e x) \sqrt {b x+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {(8 c (2 c d-b e)) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 e^4}+\frac {((4 c d-3 b e) (4 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 e^4}\\ &=\frac {2 (8 c d-3 b e+2 c e x) \sqrt {b x+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {\left (8 c (2 c d-b e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 e^4 \sqrt {b x+c x^2}}+\frac {\left ((4 c d-3 b e) (4 c d-b e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 e^4 \sqrt {b x+c x^2}}\\ &=\frac {2 (8 c d-3 b e+2 c e x) \sqrt {b x+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {\left (8 c (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left ((4 c d-3 b e) (4 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 e^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 (8 c d-3 b e+2 c e x) \sqrt {b x+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {16 \sqrt {-b} \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} (4 c d-3 b e) (4 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 1.18, size = 279, normalized size = 0.94 \[ \frac {2 (x (b+c x))^{3/2} \left (\frac {e x (b+c x) \left (c \left (8 d^2+10 d e x+e^2 x^2\right )-b e (3 d+4 e x)\right )}{d+e x}-i c e x^{3/2} \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (5 b e-8 c d) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+8 i c e x^{3/2} \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+8 (b+c x) (d+e x) (b e-2 c d)\right )}{3 e^4 x^2 (b+c x)^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 1051, normalized size = 3.53 \[ \frac {2 \sqrt {\left (c x +b \right ) x}\, \left (c^{3} e^{3} x^{4}-3 b \,c^{2} e^{3} x^{3}+10 c^{3} d \,e^{2} x^{3}-8 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} e^{3} x \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+3 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{3} e^{3} x \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+24 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c d \,e^{2} x \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-16 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c d \,e^{2} x \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-4 b^{2} c \,e^{3} x^{2}-16 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{2} d^{2} e x \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+16 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{2} d^{2} e x \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+7 b \,c^{2} d \,e^{2} x^{2}+8 c^{3} d^{2} e \,x^{2}-8 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} d \,e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+3 \sqrt {-\frac {c x}{b}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, b^{3} d \,e^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+24 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c \,d^{2} e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-16 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c \,d^{2} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-3 b^{2} c d \,e^{2} x -16 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{2} d^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+16 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{2} d^{3} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+8 b \,c^{2} d^{2} e x \right )}{3 \left (c x +b \right ) \left (e x +d \right )^{\frac {3}{2}} c \,e^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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