Optimal. Leaf size=398 \[ \frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{15 d^2 e \sqrt {d+e x} (c d-b e)^2}-\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}+\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d e^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 c d-b e)}{15 d e (d+e x)^{3/2} (c d-b e)}-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.49, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {732, 834, 843, 715, 112, 110, 117, 116} \[ \frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{15 d^2 e \sqrt {d+e x} (c d-b e)^2}-\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}+\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d e^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (2 c d-b e)}{15 d e (d+e x)^{3/2} (c d-b e)}-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 732
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx &=-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {\int \frac {b+2 c x}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx}{5 e}\\ &=-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {b x+c x^2}}{15 d e (c d-b e) (d+e x)^{3/2}}-\frac {2 \int \frac {-\frac {1}{2} b (c d-2 b e)-\frac {1}{2} c (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{15 d e (c d-b e)}\\ &=-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {b x+c x^2}}{15 d e (c d-b e) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^2 e (c d-b e)^2 \sqrt {d+e x}}+\frac {4 \int \frac {-\frac {1}{4} b c d (c d+b e)-\frac {1}{2} c \left (c^2 d^2-b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 d^2 e (c d-b e)^2}\\ &=-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {b x+c x^2}}{15 d e (c d-b e) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^2 e (c d-b e)^2 \sqrt {d+e x}}+\frac {(c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 d e^2 (c d-b e)}-\frac {\left (2 c \left (c^2 d^2-b c d e+b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{15 d^2 e^2 (c d-b e)^2}\\ &=-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {b x+c x^2}}{15 d e (c d-b e) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^2 e (c d-b e)^2 \sqrt {d+e x}}+\frac {\left (c (2 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}{15 d e^2 (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (2 c \left (c^2 d^2-b c d e+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}{15 d^2 e^2 (c d-b e)^2 \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {b x+c x^2}}{15 d e (c d-b e) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^2 e (c d-b e)^2 \sqrt {d+e x}}-\frac {\left (2 c \left (c^2 d^2-b c d e+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}{15 d^2 e^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c (2 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}{15 d e^2 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {b x+c x^2}}{15 d e (c d-b e) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{15 d^2 e (c d-b e)^2 \sqrt {d+e x}}-\frac {4 \sqrt {-b} \sqrt {c} \left (c^2 d^2-b c d e+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 )}{15 d^2 e^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} (2 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 )}{15 d e^2 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.98, size = 362, normalized size = 0.91 \[ -\frac {2 \left (b e x (b+c x) \left (-b^2 e^3 x (5 d+2 e x)+b c d e \left (-d^2+7 d e x+2 e^2 x^2\right )-c^2 d^2 \left (d^2+6 d e x+2 e^2 x^2\right )\right )+c \sqrt {\frac {b}{c}} (d+e x)^2 \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (2 b^2 e^2-3 b c d e+c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 \sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (b^2 e^2-b c d e+c^2 d^2\right )\right )\right )}{15 b d^2 e^2 \sqrt {x (b+c x)} (d+e x)^{5/2} (c d-b e)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + b x}}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 1897, normalized size = 4.77 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {c x^{2} + b x}}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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