Optimal. Leaf size=401 \[ \frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 \sqrt {c} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 \sqrt {c} e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (51 b^2 e^2-48 c e x (2 c d-b e)-176 b c d e+128 c^2 d^2\right )}{21 e^5}+\frac {10 \left (b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.48, antiderivative size = 401, 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 = {732, 812, 814, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (51 b^2 e^2-48 c e x (2 c d-b e)-176 b c d e+128 c^2 d^2\right )}{21 e^5}+\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 \sqrt {c} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{21 \sqrt {c} e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {10 \left (b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 732
Rule 812
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3 e}\\ &=\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {10 \int \frac {\left (\frac {1}{2} b (16 c d-7 b e)+8 c (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 e^3}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {4 \int \frac {-\frac {1}{4} b c d \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right )-\frac {1}{4} c (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{21 c e^5}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{21 e^6}+\frac {\left (2 d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{21 e^6}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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}{21 e^6 \sqrt {b x+c x^2}}+\frac {\left (2 d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 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}{21 e^6 \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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}{21 e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (2 d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 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}{21 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2-48 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{21 e^5}+\frac {10 (16 c d-7 b e+2 c e x) \left (b x+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 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 )}{21 \sqrt {c} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \left (128 c^2 d^2-128 b c d e+27 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 )}{21 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 2.16, size = 442, normalized size = 1.10 \[ \frac {2 (x (b+c x))^{5/2} \left (\frac {e \sqrt {x} (b+c x) \left (b^2 e^2 \left (51 d^2+67 d e x+9 e^2 x^2\right )+b c e \left (-176 d^3-224 d^2 e x-25 d e^2 x^2+9 e^3 x^3\right )+c^2 \left (128 d^4+160 d^3 e x+16 d^2 e^2 x^2-6 d e^3 x^3+3 e^4 x^4\right )\right )}{d+e x}-\frac {(b+c x) (d+e x) \left (-3 b^3 e^3+134 b^2 c d e^2-384 b c^2 d^2 e+256 c^3 d^3\right )}{c \sqrt {x}}+i e x \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-3 b^3 e^3+83 b^2 c d e^2-208 b c^2 d^2 e+128 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i e x \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (3 b^3 e^3-134 b^2 c d e^2+384 b c^2 d^2 e-256 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )}{21 e^6 x^{5/2} (b+c x)^3 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 1692, normalized size = 4.22 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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