Optimal. Leaf size=470 \[ \frac {8 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 (d+e x)^{7/2} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {8 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )}{3 b^4 c^2}+\frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b c d^2 (8 c d-11 b e)\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.59, antiderivative size = 470, 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 = {738, 818, 832, 843, 715, 112, 110, 117, 116} \[ \frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b c d^2 (8 c d-11 b e)\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )}{3 b^4 c^2}+\frac {8 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{7/2} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 738
Rule 818
Rule 832
Rule 843
Rubi steps
\begin {align*} \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {(d+e x)^{5/2} \left (\frac {1}{2} d (8 c d-11 b e)-\frac {3}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {3}{4} b d e \left (8 c^2 d^2-13 b c d e+b^2 e^2\right )+3 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c}\\ &=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {8 \int \frac {\frac {3}{8} b d e \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 {3}{8} 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 ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{9 b^4 c^2}\\ &=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^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 )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}-\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}{3 b^4 c^2}\\ &=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^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 {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 c^2 \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}{3 b^4 c^2 \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^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 {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}{3 b^4 c^2 \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}{3 b^4 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {2 \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 )}{3 (-b)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 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 )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.61, size = 451, normalized size = 0.96 \[ \frac {2 \left (b (d+e x) \left (-b c^2 d^4 (b+c x)^2+c^2 d^3 x (b+c x)^2 (8 c d-13 b e)+b x^2 (c d-b e)^4+x^2 (b+c x) (c d-b e)^3 (5 b e+8 c d)\right )-x \sqrt {\frac {b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^4 e^4+11 b^3 c d e^3+6 b^2 c^2 d^2 e^2-17 b c^3 d^3 e+8 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right )\right )\right )}{3 b^5 c^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.18, size = 2086, normalized size = 4.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________