Optimal. Leaf size=570 \[ -\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d e^6 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{63 d e^3 (d+e x)^{7/2} (c d-b e)}+\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d^2 e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{63 d^2 e^5 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}} \]
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Rubi [A] time = 0.68, antiderivative size = 570, 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, 810, 843, 715, 112, 110, 117, 116} \[ -\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{63 d e^3 (d+e x)^{7/2} (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4-320 b c^3 d^3 e+160 c^4 d^4\right )+c d^2 \left (111 b^2 c d e^2-b^3 e^3-240 b c^2 d^2 e+128 c^3 d^3\right )\right )}{63 d^2 e^5 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d e^6 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d^2 e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 732
Rule 810
Rule 843
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e}\\ &=-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {2 \int \frac {\left (-\frac {1}{2} b \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )-\frac {1}{2} c \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx}{21 d e^3 (c d-b e)}\\ &=-\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {4 \int \frac {\frac {1}{4} b c d \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+\frac {1}{2} c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{63 d^2 e^5 (c d-b e)^2}\\ &=-\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{63 d e^6 (c d-b e)}+\frac {\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{63 d^2 e^6 (c d-b e)^2}\\ &=-\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-128 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}{63 d e^6 (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-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}{63 d^2 e^6 (c d-b e)^2 \sqrt {b x+c x^2}}\\ &=-\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {\left (2 c \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-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}{63 d^2 e^6 (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) \left (128 c^2 d^2-128 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}{63 d e^6 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-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 )}{63 d^2 e^6 (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) \left (128 c^2 d^2-128 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 )}{63 d e^6 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 3.51, size = 610, normalized size = 1.07 \[ -\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (d^2 (d+e x)^2 \left (15 b^2 e^2-88 b c d e+88 c^2 d^2\right ) (c d-b e)^2-19 d^3 (d+e x) \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) (c d-b e)^2-d (d+e x)^3 \left (-b^3 e^3+63 b^2 c d e^2-183 b c^2 d^2 e+122 c^3 d^3\right ) (c d-b e)+(d+e x)^4 \left (-2 b^4 e^4-14 b^3 c d e^3+207 b^2 c^2 d^2 e^2-386 b c^3 d^3 e+193 c^4 d^4\right )+7 d^4 (c d-b e)^4\right )-c \sqrt {\frac {b}{c}} (d+e x)^4 \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-2 b^4 e^4-13 b^3 c d e^3+159 b^2 c^2 d^2 e^2-272 b c^3 d^3 e+128 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-2 \sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (b^4 e^4+7 b^3 c d e^3-135 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right )\right )\right )}{63 b d^2 e^6 x^3 (b+c x)^3 (d+e x)^{9/2} (c d-b e)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{6} x^{6} + 6 \, d e^{5} x^{5} + 15 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x + d^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 5005, normalized size = 8.78 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]
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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