Optimal. Leaf size=457 \[ \frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {2 \sqrt {-b} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Rubi [A]
time = 0.38, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {748, 846, 828,
857, 729, 113, 111, 118, 117} \begin {gather*} \frac {8 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \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} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{315 c^2 e^3}+\frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{21 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 748
Rule 828
Rule 846
Rule 857
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (b x+c x^2\right )^{3/2} \, dx &=\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {\int \sqrt {d+e x} (b d+(2 c d-b e) x) \sqrt {b x+c x^2} \, dx}{3 e}\\ &=-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {2 \int \frac {\left (\frac {1}{2} b d (c d+3 b e)+\left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{21 c e}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac {4 \int \frac {-\frac {1}{4} b d \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3\right )-\frac {1}{4} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{315 c^2 e^3}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{315 c^2 e^4}-\frac {\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{315 c^2 e^4}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-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}{315 c^2 e^4 \sqrt {b x+c x^2}}-\frac {\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{315 c^2 e^4 \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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}{315 c^2 e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-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}{315 c^2 e^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{315 c^2 e^3}-\frac {2 (2 c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac {2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac {2 \sqrt {-b} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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 )}{315 c^{5/2} e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-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 )}{315 c^{5/2} e^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 16.01, size = 463, normalized size = 1.01 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (-4 b^3 e^3+3 b^2 c e^2 (d+e x)+b c^2 e \left (-15 d^2+11 d e x+50 e^2 x^2\right )+c^3 \left (8 d^3-6 d^2 e x+5 d e^2 x^2+35 e^3 x^3\right )\right )-\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) (b+c x) (d+e x)+i b e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^4 d^4-17 b c^3 d^3 e+6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{315 b c^2 e^4 x^2 (b+c x)^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1169\) vs.
\(2(397)=794\).
time = 0.44, size = 1170, normalized size = 2.56 Too large to display
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.93, size = 516, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left ({\left (16 \, c^{5} d^{5} - 40 \, b c^{4} d^{4} e + 22 \, b^{2} c^{3} d^{3} e^{2} + 7 \, b^{3} c^{2} d^{2} e^{3} + 11 \, b^{4} c d e^{4} - 8 \, b^{5} e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 9 \, b^{2} c^{3} d^{2} e^{3} + 7 \, b^{3} c^{2} d e^{4} - 8 \, b^{4} c e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, c^{5} d^{3} e^{2} + {\left (35 \, c^{5} x^{3} + 50 \, b c^{4} x^{2} + 3 \, b^{2} c^{3} x - 4 \, b^{3} c^{2}\right )} e^{5} + {\left (5 \, c^{5} d x^{2} + 11 \, b c^{4} d x + 3 \, b^{2} c^{3} d\right )} e^{4} - 3 \, {\left (2 \, c^{5} d^{2} x + 5 \, b c^{4} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{945 \, c^{4}} \end {gather*}
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 \begin {gather*} \int \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (c\,x^2+b\,x\right )}^{3/2}\,\sqrt {d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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