Optimal. Leaf size=362 \[ \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{105 c^2 e}+\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c} \]
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Rubi [A] time = 0.39, antiderivative size = 362, 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 = {742, 814, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{105 c^2 e}+\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 e \left (b x+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 742
Rule 814
Rule 843
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx &=\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac {2 \int \frac {\left (\frac {1}{2} d (7 c d-3 b e)+2 e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 c}\\ &=\frac {2 \sqrt {d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac {4 \int \frac {\frac {1}{4} b d e \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right )+\frac {1}{4} e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{105 c^2 e^2}\\ &=\frac {2 \sqrt {d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{105 c^2 e^2}-\frac {\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{105 c^2 e^2}\\ &=\frac {2 \sqrt {d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{105 c^2 e^2 \sqrt {b x+c x^2}}-\frac {\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{105 c^2 e^2 \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac {\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \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}{105 c^2 e^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \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}{105 c^2 e^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{105 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac {2 \sqrt {-b} (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \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 )}{105 c^{5/2} e^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \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 )}{105 c^{5/2} e^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 1.86, size = 372, normalized size = 1.03 \[ \frac {2 \left (b e x (b+c x) (d+e x) \left (-4 b^2 e^2+3 b c e (3 d+e x)+3 c^2 \left (d^2+8 d e x+5 e^2 x^2\right )\right )-\sqrt {\frac {b}{c}} \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^3 e^3+23 b^2 c d e^2-18 b c^2 d^2 e+3 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{105 b c^2 e^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int \sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 920, normalized size = 2.54 \[ -\frac {2 \sqrt {e x +d}\, \sqrt {\left (c x +b \right ) x}\, \left (-15 c^{5} e^{4} x^{5}-18 b \,c^{4} e^{4} x^{4}-39 c^{5} d \,e^{3} x^{4}+b^{2} c^{3} e^{4} x^{3}-51 b \,c^{4} d \,e^{3} x^{3}-27 c^{5} d^{2} e^{2} x^{3}+4 b^{3} c^{2} e^{4} x^{2}-8 b^{2} c^{3} d \,e^{3} x^{2}-36 b \,c^{4} d^{2} e^{2} x^{2}-3 c^{5} d^{3} 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^{5} e^{4} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-27 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{4} c d \,e^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{4} c d \,e^{3} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+28 \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} c^{2} d^{2} e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-10 \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} c^{2} d^{2} e^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+4 b^{3} c^{2} d \,e^{3} x -15 \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^{3} d^{3} e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+12 \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^{3} d^{3} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-9 b^{2} c^{3} d^{2} e^{2} x +6 \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^{4} d^{4} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-6 \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^{4} d^{4} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-3 b \,c^{4} d^{3} e x \right )}{105 \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{4} e^{2} 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 \[ \int \sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int \sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^{3/2} \,d x \]
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 \[ \int \sqrt {x \left (b + c x\right )} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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